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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 19. f(x) = x2 4; P(2, 0)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 20. f(x) = 1/x; P(1, 1)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 21. f(x) = x3; P(l, l)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 22. f(x)=12x+1;P(0,1)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 23. f(x)=132x;P(1,15)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 24. f(x)=x1;P(2,1)Equations of tangent lines by definition (2) a. Use definition (2) (p. 129) to find the slope of the line tangent to the graph of f at P. b. Determine an equation of the tangent line at P. 25. f(x)=x+3;P(1,2)32EDerivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 27. f(x) = 8x; a = 3Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 28. f(x) = x2; a = 3Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 29. f(x) = 4x2 + 2x; a = 2Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 30. f(x) = 2x3; a = 10Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 31. f(x)=1x;a=14Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 32. f(x)=1x2;a=1Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 33. f(x)=2x+1;a=4Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 34. f(x)=3x;a=12Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 35. f(x)=1x+5;a=5Derivatives and tangent lines a. For the following functions and values of a, find f(a). b. Determine an equation of the line tangent to the graph of f at the point (a, f(a)) for the given value of a. 36. f(x)=13x1;a=2Derivative calculations Evaluate the derivative of the following functions at the given point. 43.f(t)=1t+1;a=1Derivative calculations Evaluate the derivative of the following functions at the given point. 44.f(t)=tt2;a=2Derivative calculations Evaluate the derivative of the following functions at the given point. 45.f(s)=2s1;a=25Derivative calculations Evaluate the derivative of the following functions at the given point. 46.f(r)=r2;a=3Explain why or why not Determine whether the following statements are true and give an explanation or counter example aFar linear functions, the slope of any secant line always equals the slope of any tangent line b.The slope of the secant line passing through the paints P and Q is less than the slope of the tangent line at P. c.Consider the graph of the parabola f(x)=x2. For a 0 and h 0, the secant line through (e, f(a)) and (a + h, f(a + h)) always has a greater slope than the tangent line at (a, f(a)).Interpreting the derivative Find the derivative of each at notion at the given point and interpret the physical meaning of this quantity. Include units in your answer. 48. When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t) = 3t. Find V(12).Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer. 49.An object dropped from rest falls d(t)=16t2 feet in t seconds. Find d(4).Interpreting the derivative Find the derivative of each at notion at the given point and interpret the physical meaning of this quantity. Include units in your answer. 50The gravitational force of attraction between two masses separated by a distance of x meters is inversely proportional to the square of the distance between them, which implies that the force is described by the function F(x)=k/x2, for some constant k where F(x) is measured in newtons. Find F(10), expression your answer in terms of k.Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer. 51Suppose the speed of a car approaching a stop sign is given by v(t)=(t5)2, for 0 t 5, where t is measured in seconds and v(t) is measured in meters per second Find v(3)Population of Las Vegas Let p(t) represent the population of the Las Vegas metropolitan area t years after 1970 as shown in the table and figure. a.Compute the average rate of growth of Las Vegas from 1970 to 1980. b.Explain why the average rate of growth calculated in part(a) is a good estimate of the instantaneous rate of growth of Las Vegas in 1975. c.Compute the average rate of growth of Las Vegas from 1970 to 2000. Is the average rate of growth an overestimate or an underestimate of the instantaneous rate of growth of Las Vegas in 2000? Approximate the growth rate in 2000.Owlet talons Let L(t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old as shown in the figure a.Estimate L(1.5) and state the physical meaning of this quantity. b.Estimate the value of L(a), for a 4. What does this tell you about the talon lengths on these birds?Caffeine levels Let 4(f) be the amount of caffeine (In mg) in the bloodstream t hours after a cup of coffee has been consumed (see figure) Estimate the values of A(7) and A(15), rounding answers to the nearest whole number, include units in your answers and interpret the physical meaning of these values.Let D(t) equal the number of daylight hours at a latitude of 40N, t days after January 1. Assuming D(t) is approximated by a continuous function (see figure), estimate the values of D(600) and D(170). Include units in your answers and Interpret your resultsFind the function The following limits represent the slope of a curve y = f(x) at the point (a, f(a)). Determine a possible function f and number a; then calculate the limit. 60. limx13x2+4x7x1Find the function The following limits represent the slope of a curve y = f(x) at the point (a. f(a)). Determine a possible function f and number a: then calculate the limit. 57.limx25x220x2Find the function The following limits represent the slope of a curve y = f(x) at the point (a, f(a)). Determine a possible function f and number a; then calculate the limit. 57. limx21x+113x2Find the function The following limits represent the slope of a curve y = f(x) at the point (a, f(a)). Determine a possible function f and number a; then calculate the limit. 59. limh0(2+h)416hFind the function The following limits represent the slope of a curve y = f(x) at the point (a, f(a)). Determine a possible function f and number a; then calculate the limit. 58. limh02+h2hFind the function The following limits represent the slope of a curve y = f(x) at the point (a. f(a)). Determine a possible function f and number a: then calculate the limit. 61.limh0|1+h|1hApproximating derivatives Assuming the limit exists, the definition of the derivative f(a)=limh0f(a+h)f(a)h implies that if h is small, then an approximation to f(a) is given by f(a)f(a+h)f(a)h. If h 0, then this approximation is called a forward difference quotient; if h 0, it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f at a point when f is a complicated function or when f is represented by a set of data points. 64. Let f(x)=x. a. Find the exact value of f(4). b. Show that f(4)f(4+h)f(4)h=4+h2h. c. Complete columns 2 and 5 of the following table and describe how 4+h2h behaves as h approaches 0. d. The accuracy of an approximation is measured by error=exactvalueapproximatevalue. Use the exact value of f(4) in part (a) to complete columns 3 and 6 in the table. Describe the behavior of the errors as h approaches 0.Another way to approximate derivatives is to use the centered difference quotient: f(a)f(a+h)f(ah)2h Again consider f(x)=x. a.Graph f near the point (4.2) and let h = 1/2 in the centered difference quotient. Draw the line whose slope is computed by the centered difference quotient and explain why the centered difference quotient approximates f(4). b.Use the centered difference quotient to approximate f(4) by completing the table. c.Explain why it is not necessary to use negative values of h in the table of part (b) d.Compare the accuracy of the derivative estimates in part (b) with those found in Exercise 62.64EApproximating derivatives Assuming the limit exists, the definition of the derivative f(a)=limh0f(a+h)f(a)h implies that if h is small, then an approximation to f(a) is given by f(a)f(a+h)f(a)h. If h 0, then this approximation is called a forward difference quotient; if h 0, it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f at a point when f is a complicated function or when f is represented by a set of data points. 67. The error function (denoted erf (x)) is an important function in statistics because it is related to the normal distribution. Its graph is shown in the figure, and values of erf (x) at several points are shown in the table. a. Use forward and centered difference quotients to find approximations to ddx(erf(x))|x=1. b. Given that ddx(erf(x))|x=1=2e, compute the error in the approximations in part (a).In Example 1, determine the slope of the tangent line at x = 2. Find the derivative of f(x)=x2+6x.What are some other ways to write f(3), where y = f(x)?In Example 2, do the slopes of the tangent lines increase or decrease as x increases? Explain.Express the derivative of p = q(r) in three ways.Is it true that if f(x) 0 at a point, than f(x) 0 at that point? Is it true that if f(x) 0 at a point, than f(x) 0 at that point? Explain.Verify tha1 the right-hand side of (1) equals f(x) if xa.For a given function f what does frepresent?If f(x)=3x+2, find the slope of the line tangent to the curve y = f(x) at x = 1,2, and 3.Why is the notation dydx used to represent the derivative?Give three different notations for the derivative of f with respect to x.Sketch a graph of a function f, where f(x) 0 and f'(x) 0 for all x in (0, 1).Sketch a graph of a function f, where f(x) 0 and f(x) 0 for all x in (0, 2).If f is differentiable at a, must f be continuous at a?If f is continuous at a, must f be differentiable at a?Describe the graph of f if f(0)=1 and f(x)=3, for x.Use the graph of f(x)=|x| to find f(x).Use limits to find f(x) if f(x)=7x.Use limits to find f(x) if f(x)=3x.13EThe weight w(x) (in pounds) of an Atlantic salmon can be estimated from its length x (in inches). If 33 x 48, the estimated weight is w(x) = 1.5x 35.8. Use limits to find w(x) and interpret its meaning.Matching functions with derivatives Match graphs ad of functions with graphs AC of their derivatives.Matching derivatives with functions Match graphs ad of derivative functions with possible graphs AD of the corresponding functions.Sketching derivatives Reproduce the graph of f and then sketch a graph of f on the same axes. 17.Sketching derivatives Reproduce the graph of f and then sketch a graph of f on the same axes. 18.Use the graph of f in the figure to do the following. a. Find the values of x in (2, 2) at which f is not continuous. b. Find the values of x in (2, 2) at which f is not differentiable.Use the graph of g in the figure to do the following. a. Find the values of x in (2, 2) at which g is not continuous. b. Find the values of x in (2, 2) at which g is not differentiable.Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 21.f(x)=5x+2;a=1,2Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 22.f(x)=7;a=1,2Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 23.f(x)=4x2+1;a=2,4Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 24.f(x)=x2+3x;a=1,4Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 25.f(x)=1x+1;a=12;5Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 26.f(x)=xx+2;a=1,0Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 27.f(x)=1t;a=9,14Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 28.f(w)=4w3;a=1,3Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 29.f(s)=4s3+3s;a=3,1Derivatives a.Use limits to find the derivative function f for the following functions f. b.Evaluate f(a) for the given values of a. 30.f(t)=3t4;a=2,2Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t). a.For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.) b.Determine the instantaneous velocity of the projectile at t = 1 and t = 2 seconds. 31.s(t)=16t2+100tVelocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t). a.For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.) b.Determine the instantaneous velocity of the projectile at t = 1 and t = 2 seconds. 32.s(t)=16t2+128t+192Evaluate dydx and dydx|x=2 if y=x+1x+2.Evaluate dydx and dydx|x=2 if s=11t3+t+1.Tangent lines a.Find the derivative function f for the following functions f b.Find an equation of the line tangent to the graph of f at (a. f(a)) for the given value of a. 35.f(x)=3x2+2x10;a=1Tangent lines a.Find the derivative function f for the following functions f b.Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. 36.f(x)=5x26x+1;a=2Calculating derivatives a. For the following functions, find f using the definition. b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. 49. f(x)=3x+1;a=8Calculating derivatives a. For the following functions, find f using the definition. b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. 50. f(x)x+2;a=7Calculating derivatives a. For the following functions, find f using the definition. b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. 51. f(x)=23x+1;a=1Calculating derivatives a. For the following functions, find f using the definition. b. Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. 52. f(x)=1x;a=5Power and energy Energy is the capacity to do work, and power is the rate at which energy is used or consumed. Therefore, if E(t) is the energy function for a system, then P(t) = E(t) is the power function. A unit of energy is the kilowatt-hour (1 kWh is the amount of energy needed to light ten 100-W lightbulbs for an hour); the corresponding units for power are kilowatts. The following figure shows the energy consumed by a small community over a 25-hour period. a. Estimate the power at t = 10 and t = 20 hr. Be sure to include units in your calculation. b. At what times on the interval [0, 25] is the power zero? c. At what times on the interval [0, 25] is the power a maximum?Slope of a line Consider the line f(x) = mx + b, where m and b are constants. Show that f(x) = m for all x. Interpret this result.A derivative formula a. Use the definition of the derivative to determine ddx(ax2+bx+c), where a, b, and c are constants. b. Let f(x) = 4x2 3x + 10 and use part (a) to find f(x). c. Use part (b) to find f(1).A derivative formula a. Use the definition of the derivative to determine ddx(ax+b), where a and b are constants. b. Let f(x)=5x+9 and use part (a) to find f(x). c. Use part (b) to find f(1).Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. a. At which points is the slope of the curve negative? b. At which points is the slope of the curve positive? c. Using AF, list the slopes in decreasing order. 53.Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. a. At which points is the slope of the curve negative? b. At which points is the slope of the curve positive? c. Using AF, list the slopes in decreasing order. 54.Matching functions with derivatives Match the functions ad in the first set of figures with the derivative functions AD in the next set of figuresSketching derivatives Reproduce the graph of f and then plot a graph of f on the same set of axes. 10.Sketching derivatives Reproduce the graph of f and then plot a graph of f on the same set of axes. 11.Sketching derivatives Reproduce the graph of f and then plot a graph of f on the same set of axes. 12.Graphing the derivative with asymptotes Sketch a graph of the derivative of the functions f shown in the figures. 13.Graphing the derivative with asymptotes Sketch a graph of the derivative of the functions f shown in the figures. 14.Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. a. Find the values of x in (0, 3) at which f is not continuous. b. Find the values of x in (0, 3) at which f is not differentiable. c. Sketch a graph of f.Where is the function continuous? Differentiable? Use the graph of g in the figure to do the following. a. Find the values of x in (0, 4) at which g is not continuous. b. Find the values of x in (0, 4) at which g is not differentiable. c. Sketch a graph of g.Voltage on a capacitor A capacitor is a device in an electrical circuit that stores charge. In one particular circuit, the charge on the capacitor Q varies in time as shown in the figure. a. At what time is the rate of change of the charge Q the greatest? b. Is Q positive or negative for t 0? c. Is Q an increasing or decreasing function of time (or neither)? d. Sketch the graph of Q. You do not need a scale on the vertical axis.Logistic growth A common model for population growth uses the logistic (or sigmoid) curve. Consider the logistic curve in the figure, where P(t) is the population at time t 0. a. At approximately what time is the rate of growth P the greatest? b. Is P positive or negative for t 0? c. Is P an increasing or decreasing function of time (or neither)? d. Sketch the graph of P You do not need a scale on the vertical axis.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the function f is differentiable for all values of x, then f is continuous for all values of x. b. The function f(x) = |x + 1| is continuous for all x, but not differentiable for all x. c. It is possible for the domain of f to be (a, b) and the domain of f to be [a, b].Looking ahead: Derivative of xn Use the definition f(x)=limh0f(x+h)f(x)h to find f(x) for the following functions. a. f(x) = x2 b. f(x) = x3 c. f(x) = x4 d. Based on your answers to parts (a)(c), propose a formula for f(x) if f(x) = xn, where n is a positive integer.59E60EFinding f from f Sketch the graph of f(x) = x. Then sketch a possible graph of f. Is more than one graph possible?62ENormal lines A line perpendicular to another line or to a tangent line is called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curses at the given point P. 21. y = 3x 4; P(1, 1)Normal lines A line perpendicular to another line or to a tangent line is called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curses at the given point P. 22. y=x;P(4,2)Normal lines A line perpendicular to another line or to a tangent line is called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curses at the given point P. 23. y=2x;P(1,2)66EAiming a tangent line Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines. 25. f(x) = x2 + 1; Q(3, 6)Aiming a tangent line Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines. 26. f(x) = x2 + 4x 3; Q(0, 6)69EAiming a tangent line Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines 70.f(x)=34x+1;Q(0,5)One-sided derivatives The right-sided and left-sided derivatives of a function at a point a are given by f+(a)=limh0+f(a+h)f(a)handf(a)=limh0f(a+h)f(a)h, respectively, provided these limits exist. The derivative f(a) exists if and only if f+(a) = f(a). a. Sketch the following functions. b. Compute f+(a) and f(a) at the given point a. c. Is f continuous at a? Is f differentiable at a? 31. f(x) = |x 2|; a = 2One-sided derivatives The right-sided and left-sided derivatives of a function at a point a are given by f+(a)=limh0+f(a+h)f(a)handf(a)=limh0f(a+h)f(a)h, respectively, provided these limits exist. The derivative f(a) exists if and only if f+(a) = f(a). a. Sketch the following functions. b. Compute f+(a) and f(a) at the given point a. c. Is f continuous at a? Is f differentiable at a? 32. f(x)={4x2ifx12x+1ifx1;a=173E74E75EVertical tangent lines If a function f is continuous at a and limxaf(x)=, then the curse y = f(x) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 3132) is used. Use this information to answer the following questions. 36. Graph the following curves and determine the location of any vertical tangent lines. a. x2 + y2 = 9 b. x2 + y2 + 2x = 0Continuity is necessary for differentiability a. Graph the function f(x)={xforx0x+1forx0.. b. For x 0, what is f(x)? c. For x 0, what is f(x)? d. Graph f on its domain. e. Is f differentiable at 0? Explain.78EFind the values of ddx(11) and ddx()Use the graph of y = x to give a geometric explanation of why ddx(x)=1.3QCFind the derivative of f(x)=4ex3x2.Determine the point(s) at which f(x) = x3 12x has a horizontal tangent line.6QCAssume the derivatives of f and g exist in Exercises 16. 1. If the limit definition of a derivative can be used to find f, then what is the purpose of using other rules to find f?Assume the derivatives of f and g exist in Exercises 16. 2. In this section, we showed that the rule ddx(xn)=nxn1 is valid for what values of n?Assume the derivatives of f and g exist in Exercises 16. 3. Give a nonzero function that is its own derivative.4EAssume the derivatives of f and g exist in Exercises 16. 5. How do you find the derivative of a constant multiplied by a function?6EGiven that f(3) = 6 and g(3) = 2, find (f + g)(3).If f(0)=6 and g(x)=f(x)+ex+1, find g(0).Let F(x)=f(x)+g(x),G(x)=f(x)g(x), and H(x)=3f(x)+2g(x), where the graphs of f and g are shown in the figure. Find each of the following. 9.F(2)Let F(x)=f(x)+g(x),G(x)=f(x)g(x), and H(x)=3f(x)+2g(x), where the graphs of f and g are shown in the figure. Find each of the following. 10.G(6)Let F(x)=f(x)+g(x),G(x)=f(x)g(x), and H(x)=3f(x)+2g(x), where the graphs of f and g are shown in the figure. Find each of the following. 11.H(2)Derivatives from a table Use the table to find the following derivatives. 58. ddx(f(x)+g(x))|x=1Derivatives from a table Use the table to find the following derivatives. 59. ddx(1.5f(x))|x=2Derivatives from a table Use the table to find the following derivatives. 60. ddx(2x3g(x))|x=4If f(t)=t10, find f(t),f(t), and f(t).Find an equation of the line tangent to the graph of y = ex at x = 0.The line tangent to the graph of f at x = 5 is y=110x2. Find ddx(4f(x))|x=5The line tangent to the graph of f at x = 3 is y = 4x 2 and the line tangent to the graph of g at x=3 is y=5x+1. Find the values of (f + g)(3) and (f + g) (3).Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 19y=x5Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 20f(t)=tDerivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 21f(x)=5Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 22g(x)=e3Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 23f(x)=5x3Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 24g(w)=56w12Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 25h(t)=t22+1Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 26f(v)=v100+ev+10Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 27p(x)=8xDerivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 28g(t)=6tDerivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 29g(t)=100t2Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 30f(s)=s4Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 31f(x)=3x4+7xDerivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 32g(x)=6x552x2+x+5Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 33f(x)=10x432x+e2Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 34f(t)=6t4t3+9Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 35g(w)=2w3+3w2+10wDerivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 36s(t)=4t14t4+t+1Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 37f(x)=3ex+5x+5Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 38g(w)=ewe2+8Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 39f(x)={x2+1ifx02x2+x+1ifx0Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of x. 40g(w)={w+5ewifw12w3+4w+5ifw1Height estimate The distance an object falls (when released from rest, under the influence of Earth s gravity and with no air resistance) is given by d(t) = 16t2, where d is measured in feet and t is measured in seconds A rock climber sits on a edge on a vertical wall and carefully observes the time it takes for a small stone to fall from the ledge to the ground. a. Compute d(t) What units are associated with the derivative and what does it measure? b. If it takes 6 s for a stone to fall to the ground, how high is the ledge? How fast is the stone moving when it strikes me ground (in miles per hour)?Projectile trajectory The position of a small rocket that is launched vertically upward is given by s(t) = 5t2 + 40t + 100, for 0 t 10, where t is measured in seconds and s is measured in meters above the ground. a. Find the rate of change in the position (instantaneous velocity) of the rocket, for 0 t 10. b. At what time is the instantaneous velocity zero? c. At what time does the instantaneous velocity have the greatest magnitude, for 0 t 10? d. Graph the position and instantaneous velocity, for 0 t 10.City urbanization City planners model the size of their city using the function A(t)=150t2+2t+20. for 0 t 50, where A is measured in square miles and t is the number of years after 2010. a. Compute A(t). What units are associated with this derivative and what does the derivative measure? b. How fast will the city be growing when it reaches a size of 38 mi2? c. Suppose that the population density of the city remains constant from year to year at 1000 people/mi2. Determine the growth rate of the population in 2030.Cell growth When observations begin at t = 0, a cell culture has 1200 cells and continues to grow according to the function p(t) = 1200 et, where p is the number of cells and t is measured in days. a. Compute p(t). What units are associated with the derivative and what does it measure? b. On the interval [0, 4], when is the growth rate p(t) the least? When is it the greatest?Weight of Atlantic salmon The weight w(x) (in pounds) of an Atlantic salmon that is x inches long can be estimated by the function w(x)={0.4x5if19x210.8x13.4if21x321.5x35.8ifx32 Calculate w(x) and explain the physical nearing of this derivativeDerivatives of products and quotients Find the derivative of the following functions by first expanding or simplify the expression. Simplify your answers. 46.f(x)=(x+1)(x1)Derivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 25. f(x) = (2x + 1)(3x2 + 2)Derivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 26. g(r) = (5r3 + 3r + 1)(r2 + 3)Derivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 29. f(w)=w3wwDerivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 30. y=12s38s2+12s4sDerivatives of products and quotients Find the derivative of the following functions by first expanding or simplify the expression. Simplify your answers. 51.h(x)=(x2+1)2Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplify the expression. Simplify your answers. 52.h(x)=x(xx3/2)Derivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 31. g(x)=x21x1Derivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 32. h(x)=x36x2+8xx22xDerivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 33. y=xaxa; a is a positive constant.Derivatives of products Find the derivative of the following functions by first expanding the expression. Simplify your answers. 34. y=x22ax+a2xa; a is a constant.Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplify the expression. Simplify your answers. 57.g(w)=e2w+ewewDerivatives of products and quotients Find the derivative of the following functions by first expanding or simplify the expression. Simplify your answers. 58.r(t)=e2t+3et+2et+2Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. 35. y = 3x2 + 2; a = 1Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. 36. y = x3 4x2 + 2x 1; a =2Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. 37. y = ex; a = ln 3Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. 38. y=ex4x; a = 0Finding slope locations Let f(x) = x3 6x + 5. a. Find the values of x for which the slope of the curve y = f(x) is 0. b. Find the values of x for which the slope of the curve y = f(x) is 2.Finding slope locations Let f(t) = t3 27t + 5. a. Find the values if t for which the slope of the curve y = f(t) is 0 b. Find the values of t for which the slope of the curve y = f(t) is 21.Finding slope locations Let f(x) = 2x3 3x2 12x + 4. a. Find all points on the graph of f at which the tangent line is horizontal. b. Find all points on the graph of f at which the tangent line has slope 60.Finding slope locations Let f(x) = 2ex 6x. a. Find all points on the graph of f at which the tangent line is horizontal. b. Find all points on the graph of f at which the tangent line has slope 12.Finding slope locations Let f(x)=4xx. a. Find all points on the graph of f at which the tangent line is horizontal. b. Find all points on the graph of f at which the tangent line has slope 12.Higher-order derivatives Find f(x), f(x), and f(x) for the following functions. 44. f(x) = 3x3 + 5x2 + 6xHigher-order derivatives Find f(x), f(x), and f(x) for the following functions. 45. f(x) = 5x4 + 10x3 + 3x +6Higher-order derivatives Find f(x), f(x), and f(x) for the following functions. 46. f(x) = 3x2 + 5exHigher-order derivatives Find f(x), f(x), and f(x) for the following functions. 47. f(x)=x27x8x+1Higher-order derivatives Find f(x), f(x), and f(x) for the following functions. 48. f(x) = 10exExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. ddx(105)=5104. b. The slope of a line tangent to f(x) = ex is never 0. c. ddx(e3)=e3. d. ddx(ex)=xex1. e. dndxn(5x3+2x+5)=0, for any integer n 3.Tangent lines Suppose f(3) = 1 and f(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x). a. Find an equation of the line tangent to y = g(x) at x = 3. b. Find an equation of the line tangent to y = h(x) at x = 3.Derivatives from tangent lines Suppose the line tangent to the graph of f at x = 2 is y = 4x + 1 and suppose the line tangent to the graph of g at x = 2 has slope 3 and passes through (0, 2). Find an equation of the line tangent to the following curves at x = 2. a. y = f(x) + g(x) b. y = f(x) 2g(x) c. y = 4f(x)Tangent line Find the equation of the line tangent to the curve y=x+x that has slope 2.Tangent line given Determine the constants b and c such that the line tangent to f(x) = x2 + bx + c at x = 1 is y = 4x + 2.Derivatives from a graph Let F = f + g and G = 3f g, where the graphs of f and g are shown in the figure. Find the following derivatives. 54. F(2)Derivatives from a graph Let F = f + g and G = 3f g, where the graphs of f and g are shown in the figure. Find the following derivatives. 55. G(2)Derivatives from a graph Let F = f + g and G = 3f g, where the graphs of f and g are shown in the figure. Find the following derivatives. 56. F(5)Derivatives from a graph Let F = f + g and G = 3f g, where the graphs of f and g are shown in the figure. Find the following derivatives. 57. G(5)Derivatives from limits The following limits represent f(a) for some function f and some real number a. a.Find a possible function f and number a. b.Evaluate the limit by computing f'(a). 82.limx0ex1xDerivatives from limits The following limits represent f(a) for some function f and some real number a. a.Find a possible function f and number a. b.Evaluate the limit by computing f'(a). 83.limx0x+ex1xDerivatives from limits The following limits represent f (a) for some function f and some real number a. a. Find a possible function f and number a. b. Evaluate the limit by computing f (a). 63. limx1x1001x1Derivatives from limits The following limits represent f (a) for some function f and some real number a. a. Find a possible function f and number a. b. Evaluate the limit by computing f (a). 61. limh09+h9hDerivatives from limits The following limits represent f (a) for some function f and some real number a. a. Find a possible function f and number a. b. Evaluate the limit by computing f (a). 62. limh(1+h)8+(1+h)32hDerivatives from limits The following limits represent f(a) for some function f and some real number a. a.Find a possible function f and number a. b.Evaluate the limit by computing f'(a). 87.limh0e3+he3h88E89ECalculator limits Use a calculator to approximate the following limits. 66. limn(1+1n)n91EConstant Rule proof For the constant function f(x) = c, use the definition of the derivative to show that f (x) = 0.93ELooking ahead: Power Rule for negative integers Suppose n is a negative integer and f(x) = xn. Use the following steps to prove that f (a) = nanl, which means the Power Rule for positive integers extends to all integers. This result is proved in Section 3.4 by a different method. a. Assume that m = n, so that m 0. Use the definition f(a)=limxaxnanxa=limxaxmamxa. Simplify using the factoring rule (which is valid for n 0) xnan=(xa)(xn1+xn2a++xan2+an1) until it is possible to take the limit. b. Use this result to find ddx(x7) and ddx(1x10).95EComputing the derivative of f(x) = ex a. Use the definition of the derivative to show that ddx(ex)=exlimh0eh1h. b. Show that the limit in part (a) is equal to 1. (Hint: Use the facts that limh0eh1h=1 and ex is continuous for all x.) c. Use parts (a) and (b) to find the derivative of f(x) = ex.97EComputing the derivative of f(x) = x2ex a. Use the definition of the derivative to show that ddx(x2ex)=exlimh0(x2+2xh+h2)ehx2h. b. Manipulate the limit in pan (a) to arrive at f(x) = ex(x2 + 2x). (Hint: Use the fact that limh0eh1h=1.)Find the derivative of f(x) = x5. Then find the same derivative using the Product Rule with f(x) = x2x3.Find the derivative of f(x)=x5 Then find the same derivative using the Quotient Rule with f(x)=x8/x3.Find the derivative of f(x)=1/x5 in two different ways: using the Power Rule and using the Quotient Rule.How do you find the derivative of the product of two functions that are differentiable at a point?How do you find the derivative of the quotient of two functions that are differentiable at a point?Use the Product Rule to evaluate and simplify ddx((x+1)(3x+2)).Use the Product Rule to find f(1) given that f(x)=x4exUse the Quotient Rule to evaluate and simplify ddx(x13x+2)Use the Quotient Rule to find g(1) given that g(x)=x2x+1Find the derivative the following ways: a. Using the Product Rule (Exercises 7-10) or the Quotient Rule (Exercises 11 -14). Simplify your result. b. By expanding the product First (Exercises 7 -10) or by simplifying the quotient first (Exercises 11-14). Verify that your answer agrees with part (a). 7.f(x)=x(x1)Find the derivative the following ways: a. Using the Product Rule (Exercises 7-10) or the Quotient Rule (Exercises 11 -14). Simplify your result. b. By expanding the product First (Exercises 7 -10) or by simplifying the quotient first (Exercises 11 -14). Verify that your answer agrees with part (a). 8. g(t)=(t+1)(t2t+1)Find the derivative the following ways: a. Using the Product Rule (Exercises 7 -10) or the Quotient Rule (Exercises 11 -14). Simplify your result. b. By expanding the product First (Exercises 7 -10) or by simplifying the quotient first (Exercises 11 -14). Verify that your answer agrees with part (a). 9.f(x)=(xl)(3x+4)Derivatives by two different methods a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part (a). 18. h(z) = (z3 + 4z2 + z)(z 1)Derivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a). 29. f(w)=w3wwFind the derivative the following ways: a. Using the Product Rule (Exercises 7 -10) or the Quotient Rule (Exercises 11 -14). Simplify your result. b. By expanding the product First (Exercises 7 -10) or by simplifying the quotient first (Exercises 11 -14). Verify that your answer agrees with part (a). 12. g(s)=4s38s2+4s4sDerivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a). 31. y=x2a2xa, where a is a constant.Derivatives by two different methods a. Use the Quotient Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by first simplifying the function. Verify that your answer agrees with part (a). 32. y=x22ax+a2xa, where a is a constant.Given that f(1)=5,f(1)=4,g(1)=2, and g(1)=3, find ddx(f(x)g(x))|x1 and ddx(f(x)g(x))|x1Show two ways to differentiate f(x) = 1/x10.Find the slope of the line tangent to the graph of f(x)=xx+6 at the point (3,1/3) and at (2,1/2).Find the slope of the graph of f(x)=2+xex at the point(0,2).Derivatives of products Find the derivative of the following functions. 7. f(x) = 3x4(2x2 1)Derivatives of products Find the derivative of the following functions. 8. g(x) = 6x 2xexDerivatives of quotients Find the derivative of the following functions. 19. f(x)=xx+1Derivatives of quotients Find the derivative of the following functions. 20. f(x)=x34x2+xx2Derivatives Find and simplify the derivative of the following functions. 23. f(t)=t5/3etDerivatives of products Find the derivative of the following functions. 10. g(w) = ew(5w2 + 3w + 1)Derivatives of quotients Find the derivative of the following functions. 21. f(x)=exex+1Derivatives of quotients Find the derivative of the following functions. 22. f(x)=2ex12ex+1Derivatives of quotients Find the derivative of the following functions. 23. f(x) = xexDerivatives Find and simplify the derivative of the following functions. 28.f(x)=exx3Derivatives of quotients Find the derivative of the following functions. 25. y = (3t 1)(2t 2)1Derivatives of quotients Find the derivative of the following functions. 26. h(w)=w21w2+1Derivatives of products Find the derivative of the following functions. 11. h(x) = (x 1)(x3 + x2 + x + 1)Derivatives of products Find the derivative of the following functions. 12. f(x)=(1+1x2)(x2+1)Derivatives of products Find the derivative of the following functions. 13. g(w) = ew(w3 1)Derivatives Find and simplify the derivative of the following functions. 34. s(t)=t4/3etDerivatives Find and simplify the derivative of the following functions. 35. f(t)=et(t22t+2)Derivatives Find and simplify the derivative of the following functions. 36. f(x)=ex(x33x2+6x6)Derivatives of quotients Find the derivative of the following functions. 27. g(x)=exx21Derivatives of quotients Find the derivative of the following functions. 28. y=(2x1)(4x+1)1Extended Power Rule Find the derivative of the following functions. 37. f(x) = 3x9Extended Power Rule Find the derivative of the following functions. 38. y=4p3Extended Power Rule Find the derivative of the following functions. 39. g(t)=3t2+6t7Extended Power Rule Find the derivative of the following functions. 40. y=w4+5w2+ww2Extended Power Rule Find the derivative of the following functions. 41. g(t)=t3+3t2+tt3Extended Power Rule Find the derivative of the following functions. 42. p(x)=4x3+3x+12x5Combining rules Compute the derivative of the following functions. 57. g(x)=(x+1)exx2Combining rules Compute the derivative of the following functions. 58. h(x)=(x1)(2x21)x31Combining rules Compute the derivative of the following functions. 59. h(x)=xexx+1Combining rules Compute the derivative of the following functions. 60. h(x)=x+1x2exDerivatives Find and simplify the derivative of the following functions. 49. g(w)=w+wwwChoose your method Use any method to evaluate the derivative of the following functions. 66. f(x)=4x2x2Derivatives Find and simplify the derivative of the following functions. 51.h(w)=w5/3w5/3+1Derivatives Find and simplify the derivative of the following functions. 52.g(x)=x4/31x4/3+1Choose your method Use any method to evaluate the derivative of the following functions. 67. f(x)=4x22x5x+1Derivatives Find and simplify the derivative of the following functions. 54. f(z)=(z2+1z)ezChoose your method Use any method to evaluate the derivative of the following functions. 69. h(r)=2rrr+1Choose your method Use any method to evaluate the derivative of the following functions. 70. y=xaxa, where a is a positive constant.Choose your method Use any method to evaluate the derivative of the following functions. 71. h(x) = (5x7 + 5x)(6x3 + 3x2 + 3)Derivatives Find and simplify the derivative of the following functions. 58. s(t)(t+1)(t+2)(t+3)Derivatives Find and simplify the derivative of the following functions. 59.f(x)=e2x+8x2ex+16x4 (Hint: Factor the function under the square root first.)Derivatives Find and simplify the derivative of the following functions. 60. g(x)=e2x1ex1Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent tine on the same set of axes. 33. y=x+5x1; a = 3Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent tine on the same set of axes. 34. y=2x23x1; a = 1Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent tine on the same set of axes. 35. y = 1 + 2x + xex; a = 0Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent tine on the same set of axes. 36. y=exx; a = 1Population growth Consider the following population functions. a. Find the instantaneous growth rate of the population, for t 0. b. What is the instantaneous growth rate at t = 5? c. Estimate the lime when the instantaneous growth rate is the greatest. d. Evaluate and interpret limtp(t). e. Use a graphing utility to graph the population and its growth rate. 51. p(t)=200tt+2Population growth Consider the following population functions. a. Find the instantaneous growth rate of the population, for t0. b. What is the instantaneous growth rate at t=5? c. Estimate the time when the instantaneous growth rate is greatest. d. Evaluate and interpret limtp(t) e. Use a graphing utility to graph the population and its growth rate. 66.p(t)=600(t2+3t2+9)Electrostatic force The magnitude of the electrostatic force between two point charges Q and q of the same sign is given by F(x)=kQqx2, where x is the distance (measured in meters) between the charges and k = 9 109 Nm2/C2 is a physical constant (C stands for coulomb, the unit of charge: N stands for newton, the unit of force). a. Find the instantaneous rate of change of the force with respect to the distance between the charges. b. For two identical charges with Q = q = 1 C, what is the instantaneous rate of change of the force at a separation of x = 0.001 m? c. Does the magnitude of the instantaneous rate of change of the force increase or decrease with the separation? Explain.Gravitational force The magnitude of the gravitational force between two objects of mass M and m is given by F(x)=GMmx2, where x is the distance between the centers of mass of the objects and G = 6.7 1011 Nm2/kg2 is the gravitational constant (N stands for newton, the unit of force: the negative sign indicates an attractive force). a. Find the instantaneous rate of change of the force with respect to the distance between the objects. b. For two identical objects of mass M = m = 0.1 kg, what is the instantaneous rate of change of the force at a separation of x = 0.01 m? c. Does the instantaneous rate of change of the force increase or decrease with the separation? Explain.