Calculus: Early Transcendentals, Enhanced Etext
12th Edition
ISBN: 9781119777984
Author: Howard Anton; Irl C. Bivens; Stephen Davis
Publisher: Wiley Global Education US
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Chapter 3.1, Problem 12ES
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Question 1. (10 points)
A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by
dV
=
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dt
(a) (4 pts) Find all the equilibria and determine their stability using the stability condition.
(b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of
f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable.
(c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain
in biological terms what happens to the size of each of these tumours at time progresses.
For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.
Question 2: (10 points) Evaluate the definite integral.
Use the following form of the definition of the integral to evaluate the integral:
Theorem: Iff is integrable on [a, b], then
where Ax = (ba)/n and x₂ = a + i^x.
You might need the following formulas.
IM³
L² (3x²
(3x²+2x-
2x - 1)dx.
n
[f(z)dz lim f(x)Az
a
n→∞
i=1
n(n + 1)
2
n
i=1
n(n+1)(2n+1)
6
Chapter 3 Solutions
Calculus: Early Transcendentals, Enhanced Etext
Ch. 3.1 - The equation xy+2y=1 defines implicitly the...Ch. 3.1 - Use implicit differentiation to find dy/dx for...Ch. 3.1 - The slope of the tangent line to the graph of...Ch. 3.1 - Use implicit differentiation to find d2y/dx2 for...Ch. 3.1 - (a) Find dy/dx by differentiating implicitly. (b)...Ch. 3.1 - (a) Find dy/dx by differentiating implicitly. (b)...Ch. 3.1 - Prob. 3ESCh. 3.1 - Find dy/dx by implicit differentiation. x3+y3=3xy2Ch. 3.1 - Find dy/dx by implicit differentiation....Ch. 3.1 - Find dy/dx by implicit differentiation....
Ch. 3.1 - Find dy/dx by implicit differentiation. 1x+1y=1Ch. 3.1 - Prob. 8ESCh. 3.1 - Find dy/dx by implicit differentiation. sinx2y2=xCh. 3.1 - Prob. 10ESCh. 3.1 - Find dy/dx by implicit differentiation....Ch. 3.1 - Prob. 12ESCh. 3.1 - Find d2y/dx2 by implicit differentiation. 2x23y2=4Ch. 3.1 - Prob. 14ESCh. 3.1 - Find d2y/dx2 by implicit differentiation. x3y34=0Ch. 3.1 - Prob. 16ESCh. 3.1 - Find d2y/dx2 by implicit differentiation. y+siny=xCh. 3.1 - Find d2y/dx2 by implicit differentiation. xcosy=yCh. 3.1 - In each part, refer to the equation x+y=xy. a....Ch. 3.1 - In each part, refer to the equation x2=x+yxy a....Ch. 3.1 - In the accompanying figure the area of the blue...Ch. 3.1 - True-False Determine whether the statement is true...Ch. 3.1 - True-False Determine whether the statement is true...Ch. 3.1 - True-False Determine whether the statement is true...Ch. 3.1 - True-False Determine whether the statement is true...Ch. 3.1 - Use implicit differentiation to find the slope of...Ch. 3.1 - Use implicit differentiation to find the slope of...Ch. 3.1 - Use implicit differentiation to find the slope of...Ch. 3.1 - Use implicit differentiation to find the slope of...Ch. 3.1 - Use implicit differentiation to find the specified...Ch. 3.1 - Use implicit differentiation to find the specified...Ch. 3.1 - Use implicit differentiation to find the specified...Ch. 3.1 - Prob. 34ESCh. 3.1 - As shown in the accompanying figure, it appears...Ch. 3.1 - (a) A student claims that the ellipse x2xy+y2=1...Ch. 3.1 - (a) Use the implicit plotting capability of a CAS...Ch. 3.1 - Use implicit differentiation to find all points on...Ch. 3.1 - Find the values of a and b for the curve x2y+ay2=b...Ch. 3.1 - At what point(s) is the tangent line to the curve...Ch. 3.1 - Two curves are said to be orthogonal if their...Ch. 3.1 - Two curves are said to be orthogonal if their...Ch. 3.1 - (a) Use the implicit plotting capability of a CAS...Ch. 3.1 - (a) Use the implicit plotting capability of a CAS...Ch. 3.1 - Find dy/dx if 2y3t+t3y=1 and dtdx=1costCh. 3.1 - Find equations for two lines through the origin...Ch. 3.1 - A student asks: “Suppose implicit...Ch. 3.2 - The equation of the tangent line to the graph of...Ch. 3.2 - Find dy/dx . (a) y=ln3x (b) y=lnx (c) y=log1/xCh. 3.2 - Use logarithmic differentiation to find the...Ch. 3.2 - limh0ln1+hh=Ch. 3.2 - Find dy/dx . y=ln5xCh. 3.2 - Prob. 2ESCh. 3.2 - Find dy/dx . y=ln1+xCh. 3.2 - Prob. 4ESCh. 3.2 - Find dy/dx . y=lnx21Ch. 3.2 - Find dy/dx . y=lnx37x23Ch. 3.2 - Find .
Ch. 3.2 - Prob. 8ESCh. 3.2 - Find .
Ch. 3.2 - Find dy/dx . y=lnx3Ch. 3.2 - Find dy/dx . y=lnxCh. 3.2 - Find .
Ch. 3.2 - Find dy/dx 13.y=x2lnxCh. 3.2 - Find .
Ch. 3.2 - Find dy/dx . y=x2log232xCh. 3.2 - Find dy/dx . y=xlog2x22x3Ch. 3.2 - Find dy/dx . y=x21+logxCh. 3.2 - Find dy/dx . y=logx1+logxCh. 3.2 - Find dy/dx . y=lnlnxCh. 3.2 - Prob. 20ESCh. 3.2 - Find dy/dx . y=lntanxCh. 3.2 - Find dy/dx . y=lncosxCh. 3.2 - Find dy/dx . y=coslnxCh. 3.2 - Find dy/dx . y=sin2lnxCh. 3.2 - Find dy/dx . y=logsin2xCh. 3.2 - Prob. 26ESCh. 3.2 - Use the method of Example 3 to help perform the...Ch. 3.2 - Use the method of Example 3 to help perform the...Ch. 3.2 - Use the method of Example 3 to help perform the...Ch. 3.2 - Use the method of Example 3 to help perform the...Ch. 3.2 - Determine whether the statement is true or false....Ch. 3.2 - Determine whether the statement is true or false....Ch. 3.2 - Determine whether the statement is true or false....Ch. 3.2 - Determine whether the statement is true or false....Ch. 3.2 - Find dy/dx using logarithmic differentiation....Ch. 3.2 - Find dy/dx using logarithmic differentiation....Ch. 3.2 - Find dy/dx using logarithmic differentiation....Ch. 3.2 - Find dy/dx using logarithmic differentiation....Ch. 3.2 - Find (a) ddxlogxe (b) ddxlogx2.Ch. 3.2 - Find (a) ddxlog1/xe (b) ddxloglnxe.Ch. 3.2 - Find the equation of the tangent line to the graph...Ch. 3.2 - Find the equation of the tangent line to the graph...Ch. 3.2 - Find the equation of the tangent line to the graph...Ch. 3.2 - Find the equation of the tangent line to the graph...Ch. 3.2 - (a) Find the equation of a line through the origin...Ch. 3.2 - Use logarithmic differentiation to verify the...Ch. 3.2 - Find a formula for the area Aw of the triangle...Ch. 3.2 - Find a formula for the area Aw of the triangle...Ch. 3.2 - Verify that y=lnx+e satisfies dy/dx=ey , with y=1...Ch. 3.2 - Verify that y=lne2x satisfies dy/dx=ey , with y=2...Ch. 3.2 - Find a function 0 such that y=fx satisfies...Ch. 3.2 - Find a function f such that y=fx satisfies...Ch. 3.2 - Find the limit by interpreting the expression as...Ch. 3.2 - Find the limit by interpreting the expression as...Ch. 3.2 - Find the limit by interpreting the expression as...Ch. 3.2 - Modify the derivation of Equation (2) to give...Ch. 3.2 - Let p denote the number of paramecia in a nutrient...Ch. 3.2 - One model for the spread of information over time...Ch. 3.3 - Suppose that a one-to-one function f has tangent...Ch. 3.3 - In each case, from the given derivative, determine...Ch. 3.3 - Evaluate the derivative.
(a)
(b)
(c)
(d)
Ch. 3.3 - Let fx=ex3+x . Use fx to verify that f is...Ch. 3.3 - Let fx=x5+x3+x . (a) Show that f is one-to-one and...Ch. 3.3 - Prob. 2ESCh. 3.3 - Find f1x using Formula (2), and check your answer...Ch. 3.3 - Find f1(x) using Formula (2), and check your...Ch. 3.3 - Determine whether the function f is one-to-one on...Ch. 3.3 - Determine whether the function f is one-to-one by...Ch. 3.3 - Find the derivative of f1 by using Formula (3),...Ch. 3.3 - Prob. 8ESCh. 3.3 - Find the derivative of f1 by using Formula (3),...Ch. 3.3 - Find the derivative of f1 by using Formula (3),...Ch. 3.3 - Complete each part to establish that the...Ch. 3.3 - Prove that the reflection about the line y=x of a...Ch. 3.3 - Suppose that and are increasing functions....Ch. 3.3 - Suppose that f and g are one-to-one functions....Ch. 3.3 - Find dy/dx . y=e7xCh. 3.3 - Find dy/dx . y=e5x2Ch. 3.3 - Find dy/dx . y=x3exCh. 3.3 - Find dy/dx . y=e1/xCh. 3.3 - Find dy/dx . y=exexex+exCh. 3.3 - Find dy/dx . y=sinexCh. 3.3 - Find dy/dx . y=extanxCh. 3.3 - Find dy/dx . y=exlnxCh. 3.3 - Find dy/dx . y=exe3xCh. 3.3 - Prob. 24ESCh. 3.3 - Find dy/dx . y=ln1xexCh. 3.3 - Prob. 26ESCh. 3.3 - Find fx by Formula (7) and then by logarithmic...Ch. 3.3 - Find fx by Formula (7) and then by logarithmic...Ch. 3.3 - Find fx by Formula (7) and then by logarithmic...Ch. 3.3 - Find fx by Formula (7) and then by logarithmic...Ch. 3.3 - Find dy/dx using the method of logarithmic...Ch. 3.3 - Prob. 32ESCh. 3.3 - Find dy/dx using the method of logarithmic...Ch. 3.3 - Prob. 34ESCh. 3.3 - Find dy/dx using the method of logarithmic...Ch. 3.3 - (a) Explain why Formula (5) cannot be used to find...Ch. 3.3 - Find dy/dx using any method. y=x32x2+1exCh. 3.3 - Prob. 38ESCh. 3.3 - Find dy/dx using any method. y=x2+x3xCh. 3.3 - Prob. 40ESCh. 3.3 - Find dy/dx using any method. y=43sinxexCh. 3.3 - Prob. 42ESCh. 3.3 - Find dy/dx . y=sin13xCh. 3.3 - Prob. 44ESCh. 3.3 - Find dy/dx . y=sin11/xCh. 3.3 - Find dy/dx . y=cos1cosxCh. 3.3 - Find dy/dx . y=tan1x3Ch. 3.3 - Prob. 48ESCh. 3.3 - Find dy/dx . y=tanx1Ch. 3.3 - Find dy/dx . y=1tan1xCh. 3.3 - Find dy/dx . y=exsec1xCh. 3.3 - Find dy/dx . y=lncos1xCh. 3.3 - Find dy/dx . y=sin1x+cos1xCh. 3.3 - Find dy/dx . y=x2sin1x3Ch. 3.3 - Find dy/dx . y=sec1x+csc1xCh. 3.3 - Find dy/dx . y=csc1exCh. 3.3 - Find dy/dx . y=cot1xCh. 3.3 - Find dy/dx . y=cot1xCh. 3.3 - Determine whether the statement is true or false....Ch. 3.3 - Determine whether the statement is true or false....Ch. 3.3 - Determine whether the statement is true or false....Ch. 3.3 - Determine whether the statement is true or false....Ch. 3.3 - (a) Use Formula (2) to prove that ddxcot1xx=0=1...Ch. 3.3 - (a) Use part (c) of Exercise 30 in Section 1.7 and...Ch. 3.3 - Find dy/dx by implicit differentiation....Ch. 3.3 - Find dy/dx by implicit differentiation....Ch. 3.3 - (a) Show that fx=x33x2+2x is not one-to-one on ,+...Ch. 3.3 - (a) Show that fx=x42x3 is not one-to-one on ,+ ....Ch. 3.3 - Let fx=x4+x3+1,0x2 . (a) Show that f is...Ch. 3.3 - Let fx=exp4x2x,x0 . (a) Show that f is one-to-one....Ch. 3.3 - Show that for any constant A and k , then function...Ch. 3.3 - Show that for any constants A and B , the function...Ch. 3.3 - Show that (a) y=xex satisfies the equation xy=1xy...Ch. 3.3 - Prob. 75ESCh. 3.3 - Find the limit by interpreting the expression as...Ch. 3.3 - Find the limit by interpreting the expression as...Ch. 3.3 - Find the limit by interpreting the expression as...Ch. 3.3 - Find the limit by interpreting the expression as...Ch. 3.3 - Find the limit by interpreting the expression as...Ch. 3.3 - Find the limit by interpreting the expression as...Ch. 3.3 - Suppose that a steel ball bearing is released...Ch. 3.4 - If A=x2 and dxdt=3 , find dAdtx=10.Ch. 3.4 - If A=x2 and dAdt=3 , find dxdtx=10.Ch. 3.4 - A 10-foot ladder stands on a horizontal floor and...Ch. 3.4 - Suppose that a block of ice in the shape of a...Ch. 3.4 - Both x and y denote functions of t that are...Ch. 3.4 - Both x and y denote functions of t that are...Ch. 3.4 - Both x and y denote functions of t that are...Ch. 3.4 - Both x and y denote functions of t that are...Ch. 3.4 - Prob. 5ESCh. 3.4 - In parts (a)-(d), let A be the area of a circle of...Ch. 3.4 - Let V be the volume of a cylinder having height h...Ch. 3.4 - Prob. 8ESCh. 3.4 - Let (in radians) be an acute angle in a right...Ch. 3.4 - Suppose that z=x3y2 , where both x and y are...Ch. 3.4 - The minute hand of a certain clock is 4in long....Ch. 3.4 - A stone dropped into a still pond sends out a...Ch. 3.4 - Oil spilled from a ruptured tanker spreads in a...Ch. 3.4 - Prob. 14ESCh. 3.4 - Suppose the balloon described in the previous...Ch. 3.4 - Prob. 16ESCh. 3.4 - A 13ft ladder is leaning against a wall. If the...Ch. 3.4 - A 10ft plank is leaning against a wall. If at a...Ch. 3.4 - A softball diamond is a square whose sides are...Ch. 3.4 - A satellite is in an elliptical orbit around the...Ch. 3.4 - An aircraft is flying horizontally at a constant...Ch. 3.4 - A conical water tank with vertex down has a radius...Ch. 3.4 - Grain pouring from a chute at the rate of 8ft3/min...Ch. 3.4 - Sand pouring from a chute forms a conical pile...Ch. 3.4 - Wheat is poured through a chute at the rate of...Ch. 3.4 - An aircraft is climbing at a 30 angle to the...Ch. 3.4 - A boat is pulled into a dock by means of a rope...Ch. 3.4 - For the boat in Exercise 30, how fast must the...Ch. 3.4 - A man 6ft tall is walking at the rate of 3ft/s...Ch. 3.4 - A beacon that makes one revolution every 10s is...Ch. 3.4 - An aircraft is flying at a constant altitude with...Ch. 3.4 - Solve Exercise 34 under the assumption that the...Ch. 3.4 - A police helicopter is flying due north at 100mi/h...Ch. 3.4 - Prob. 37ESCh. 3.4 - Prob. 40ESCh. 3.4 - A particle is moving along the curve y=x/x2+1 ....Ch. 3.4 - A new design for a wind turbine adjusts the length...Ch. 3.4 - The thin lens equation in physics is 1s+1S=1f...Ch. 3.4 - Water is stored in a cone-shaped reservoir (vertex...Ch. 3.4 - A meteor enters the Earth’s atmosphere and bums...Ch. 3.4 - On a certain clock the minute hand is 4in long and...Ch. 3.4 - Coffee is poured at a uniform rate of 20cm3/s into...Ch. 3.5 - The local linear approximation of f at x0 use the ...Ch. 3.5 - Find an equation for the local linear...Ch. 3.5 - Let y=5x2 . Find dy and y at x=2 with dx=x=0.1 .Ch. 3.5 - Prob. 4QCECh. 3.5 - (a) Use Formula (1) to obtain the local linear...Ch. 3.5 - (a) Use Formula (1) to obtain the local linear...Ch. 3.5 - (a) Find the local linear approximation of the...Ch. 3.5 - A student claims that whenever a local linear...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the stated formula is the local...Ch. 3.5 - Confirm that the formula is the local linear...Ch. 3.5 - Confirm that the formula is the local linear...Ch. 3.5 - Confirm that the formula is the local linear...Ch. 3.5 - Confirm that the formula is the local linear...Ch. 3.5 - (a) Use the local linear approximation of sinx at...Ch. 3.5 - (a) Use the local linear approximation of at to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - Use an appropriate local linear approximation to...Ch. 3.5 - The approximation 1+xk1+kx is commonly used by...Ch. 3.5 - Use the approximation 1+xk1+kx , along with some...Ch. 3.5 - Referring to the accompanying figure, suppose that...Ch. 3.5 - Prob. 39ESCh. 3.5 - (a) Let y=x . Find dy and y at x=9 with dx=x=1 ....Ch. 3.5 - Find formulas for and .
Ch. 3.5 - Find formulas for and .
Ch. 3.5 - Find formulas for dy and y . y=x22x+1Ch. 3.5 - Find formulas for and .
Ch. 3.5 - Find the differential dy . (a) y=4x37x2 (b)...Ch. 3.5 - Find the differential .
(a)
(b)
Ch. 3.5 - Find the differential dy . (a) y=x1x (b) y=1+x17Ch. 3.5 - Find the differential dy . (a) y=1x31 (b) y=1x32xCh. 3.5 - Determine whether the statement is true or false....Ch. 3.5 - Determine whether the statement is true or false....Ch. 3.5 - Determine whether the statement is true or false....Ch. 3.5 - Determine whether the statement is true or false....Ch. 3.5 - Use the differential to approximate when ...Ch. 3.5 - Use the differential dy to approximate y when x...Ch. 3.5 - Use the differential dy to approximate y when x...Ch. 3.5 - Use the differential dy to approximate y when x...Ch. 3.5 - A long high-voltage power line is 18feet above the...Ch. 3.5 - A metal rod 15cm long and 5cm in diameter is to be...Ch. 3.5 - The magnitude R of an earthquake on the Richter...Ch. 3.5 - Prob. 62ESCh. 3.5 - The side of a square is measured to be 10ft , with...Ch. 3.5 - The side of a cube is measured to be 25cm , with a...Ch. 3.5 - The hypotenuse of a right triangle is known to be...Ch. 3.5 - One side of a right triangle is known to be 25cm...Ch. 3.5 - The electrical resistance R of a certain wire is...Ch. 3.5 - The side of a square is measured with a possible...Ch. 3.5 - The side of a cube is measured with a possible...Ch. 3.5 - The volume of a sphere is to be computed from a...Ch. 3.5 - The area of a circle is to be computed from a...Ch. 3.5 - The time required for one complete oscillation of...Ch. 3.5 - Suppose that the time T (in days) for a cancerous...Ch. 3.5 - Explain why the local linear approximation of a...Ch. 3.6 - Prob. 1QCECh. 3.6 - Evaluate each of the limits in Quick Check...Ch. 3.6 - Using L’Hopital’s rule, limx+ex500x2=.Ch. 3.6 - Evaluate the given limit without using...Ch. 3.6 - Evaluate the given limit without using...Ch. 3.6 - Determine whether the statement is true or false....Ch. 3.6 - Determine whether the statement is true or false....Ch. 3.6 - Determine whether the statement is true or false....Ch. 3.6 - Determine whether the statement is true or false....Ch. 3.6 - Prob. 7ESCh. 3.6 - Find the limits. limx0sin2xsin5xCh. 3.6 - Prob. 9ESCh. 3.6 - Find the limits. limt0tet1etCh. 3.6 - Prob. 11ESCh. 3.6 - Prob. 12ESCh. 3.6 - Find the limits. limx+lnxxCh. 3.6 - Find the limits. limx+e3xx2Ch. 3.6 - Find the limits. 15. limx0+cot2xlnxCh. 3.6 - Find the limits. 16. limx0+1logxe1/xCh. 3.6 - Find the limits. limx+x100exCh. 3.6 - Prob. 18ESCh. 3.6 - Find the limits. limx0sin12xxCh. 3.6 - Find the limits. limx0xtan1xx3Ch. 3.6 - Find the limits. limx+xexCh. 3.6 - Find the limits. limxxtan12xCh. 3.6 - Prob. 23ESCh. 3.6 - Find the limits. limx0+tanxlnxCh. 3.6 - Prob. 25ESCh. 3.6 - Find the limits. limxxcotxCh. 3.6 - Find the limits. limx+13/xxCh. 3.6 - Prob. 28ESCh. 3.6 - Prob. 29ESCh. 3.6 - Find the limits. limx+1+a/xbxCh. 3.6 - Prob. 31ESCh. 3.6 - Find the limits. limx+cos2/xx2Ch. 3.6 - Find the limits. limx0cscx1/xCh. 3.6 - Prob. 34ESCh. 3.6 - Find the limits. limx+x2+xxCh. 3.6 - Prob. 36ESCh. 3.6 - Find the limits. limx+xlnx2+1Ch. 3.6 - Prob. 38ESCh. 3.6 - Find the limits. limx0+xsinxCh. 3.6 - Find the limits. limx0+e2x1xCh. 3.6 - Find the limits. limx0+1lnxxCh. 3.6 - Find the limits. limx+x1/xCh. 3.6 - Find the limits. limx+lnx1/xCh. 3.6 - Find the limits. limx0+lnxxCh. 3.6 - Prob. 45ESCh. 3.6 - Show that for any positive integer n (a)...Ch. 3.6 - (a) Find the error in the following calculation:...Ch. 3.6 - (a) Find the error in the following calculation:...Ch. 3.6 - Make a conjecture about the limit by graphing the...Ch. 3.6 - Make a conjecture about the limit by graphing the...Ch. 3.6 - Make a conjecture about the limit by graphing the...Ch. 3.6 - Make a conjecture about the limit by graphing the...Ch. 3.6 - Make a conjecture about the equations of...Ch. 3.6 - Make a conjecture about the equations of...Ch. 3.6 - Make a conjecture about the equations of...Ch. 3.6 - Make a conjecture about the equations of...Ch. 3.6 - Prob. 57ESCh. 3.6 - There is a myth that circulates among beginning...Ch. 3.6 - Verify that L’Hopital’s rule is of no help in...Ch. 3.6 - Verify that L’Hopital’s rule is of no help in...Ch. 3.6 - Verify that L’Hopital’s rule is of no help in...Ch. 3.6 - Verify that L’Hopital’s rule is of no help in...Ch. 3.6 - The accompanying schematic diagram represents an...Ch. 3.6 - (a) Show that limx/2/2xtanx=1 . (b) Show that...Ch. 3.6 - (a) Use a CAS to show that if k is a positive...Ch. 3.6 - Find all values of k and l such that...Ch. 3.6 - Let fx=x2sin1/x . (a) Are the limits limx0+fx and...Ch. 3.6 - (a) Explain why L’Hopital’s rule does not...Ch. 3.6 - Find limx0+xsin1/xsinx if it exists.Ch. 3.6 - Suppose that functions f and g are differentiable...Ch. 3.6 - Were we to use L’Hopital’s mle to evaluate...Ch. 3 - (a) Find dy/dx by differentiating implicitly, (b)...Ch. 3 - (a) Find dy/dx by differentiating implicitly. (b)...Ch. 3 - Find dy/dx by implicit differentiation. 1y+1x=1Ch. 3 - Find dy/dx by implicit differentiation. x3y3=6xyCh. 3 - Find dy/dx by implicit differentiation. secxy=yCh. 3 - Prob. 6RECh. 3 - Find d2y/dx2 by implicit differentiation. 3x24y2=7Ch. 3 - Prob. 8RECh. 3 - Use implicit differentiation to find the slope of...Ch. 3 - Prob. 10RECh. 3 - Prove that if P and Q are two distinct points on...Ch. 3 - Find the coordinates of the point in the first...Ch. 3 - Find the coordinates of the point in the first...Ch. 3 - Use implicit differentiation to show that the...Ch. 3 - Find dy/dx by first using algebraic properties of...Ch. 3 - Find dy/dx by first using algebraic properties of...Ch. 3 - Find dy/dx . y=ln2xCh. 3 - Find dy/dx . y=lnx2Ch. 3 - Find dy/dx . y=lnx+13Ch. 3 - Find dy/dx . y=lnx+13Ch. 3 - Find dy/dx . y=loglnxCh. 3 - Find dy/dx . y=1+logx1logxCh. 3 - Find dy/dx . y=lnx3/21+x4Ch. 3 - Prob. 24RECh. 3 - Find dy/dx . y=elnx2+1Ch. 3 - Prob. 26RECh. 3 - Find dy/dx . y=2xexCh. 3 - Find dy/dx . y=a1+bexCh. 3 - Find dy/dx . y=1tan12xCh. 3 - Find dy/dx . y=2sin1xCh. 3 - Find dy/dx . y=xexCh. 3 - Find dy/dx . y=1+x1/xCh. 3 - Find dy/dx . y=sec12x+1Ch. 3 - Find dy/dx . y=cos1x2Ch. 3 - Find dy/dx using logarithmic differentiation....Ch. 3 - Find dy/dx using logarithmic differentiation....Ch. 3 - (a) Make a conjecture about the shape of the graph...Ch. 3 - Recall from Section 1.8 that the loudness of a...Ch. 3 - A particle is moving along the curve y=xlnx . Find...Ch. 3 - Find the equation of the tangent fine to the graph...Ch. 3 - Find the value of b so that the line y=x is...Ch. 3 - In each part, find the value of k for which the...Ch. 3 - If f and g are inverse functions and f is...Ch. 3 - In each part, find f1x using Formula (2) of...Ch. 3 - Find a point on the graph of y=e3x at which the...Ch. 3 - Prob. 46RECh. 3 - Prob. 47RECh. 3 - The equilibrium constant k of a balanced chemical...Ch. 3 - Show that the function y=eaxsinbx satisfies...Ch. 3 - Show that the function y=tan1x satisfies...Ch. 3 - Suppose that the population of deer on an island...Ch. 3 - In each part, find each limit by interpreting the...Ch. 3 - Suppose that limfx= and limgx= . In each of the...Ch. 3 - Prob. 54RECh. 3 - Evaluate the given limit. limx+exx2Ch. 3 - Evaluate the given limit. limx1lnxx41Ch. 3 - Evaluate the given limit. limx0x2e2sin23xCh. 3 - Evaluate the given limit. limx0ax1x,a0Ch. 3 - An oil slick on a lake is surrounded by a floating...Ch. 3 - If the closest approach of a planet to the Sun...Ch. 3 - In each part, use the given information to find...Ch. 3 - Use an appropriate local linear approximation to...Ch. 3 - The base of the Great Pyramid at Giza is a square...
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- For the system consisting of the three planes:plane 1: -4x + 4y - 2z = -8plane 2: 2x + 2y + 4z = 20plane 3: -2x - 3y + z = -1a) Are any of the planes parallel and/or coincident? Justify your answer.b) Determine if the normals are coplanar. What does this tell you about the system?c) Solve the system if possible. Show a complete solution (do not use matrix operations). Classify the system using the terms: consistent, inconsistent, dependent and/or independent.arrow_forwardFor the system consisting of the three planes:plane 1: -4x + 4y - 2z = -8plane 2: 2x + 2y + 4z = 20plane 3: -2x - 3y + z = -1a) Are any of the planes parallel and/or coincident? Justify your answer.b) Determine if the normals are coplanar. What does this tell you about the system?c) Solve the system if possible. Show a complete solution (do not use matrix operations). Classify the system using the terms: consistent, inconsistent, dependent and/or independent.arrow_forwardOpen your tool box and find geometric methods, symmetries of even and odd functions and the evaluation theorem. Use these to calculate the following definite integrals. Note that you should not use Riemann sums for this problem. (a) (4 pts) (b) (2 pts) 3 S³ 0 3-x+9-dz x3 + sin(x) x4 + cos(x) dx (c) (4 pts) L 1-|x|dxarrow_forward
- An engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides to do it in three sections. The first section runs from point P to point Q, and costs $48 per mile to lay, the second section runs from point Q to point R and costs $54 per mile, the third runs from point R to point S and costs $44 per mile. Looking at the diagram below, you see that if you know the lengths marked x and y, then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline. Please show your answers to 4 decimal places. 2 Miles x = 1 Mile R 10 miles miles y = milesarrow_forwardAn open-top rectangular box is being constructed to hold a volume of 150 in³. The base of the box is made from a material costing 7 cents/in². The front of the box must be decorated, and will cost 11 cents/in². The remainder of the sides will cost 3 cents/in². Find the dimensions that will minimize the cost of constructing this box. Please show your answers to at least 4 decimal places. Front width: Depth: in. in. Height: in.arrow_forwardFind and classify the critical points of z = (x² – 8x) (y² – 6y). Local maximums: Local minimums: Saddle points: - For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. Enter DNE if there are no points for a classification.arrow_forward
- Suppose that f(x, y, z) = (x − 2)² + (y – 2)² + (z − 2)² with 0 < x, y, z and x+y+z≤ 10. 1. The critical point of f(x, y, z) is at (a, b, c). Then a = b = C = 2. Absolute minimum of f(x, y, z) is and the absolute maximum isarrow_forwardThe spread of an infectious disease is often modeled using the following autonomous differential equation: dI - - BI(N − I) − MI, dt where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of transmission, and μ is the rate at which people recover from infection. Close a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria. b) (5 points) For the equilbria in part a), determine whether each is stable or unstable. c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the dt function by hand.) Identify the equilibria as stable or unstable in the graph. d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.arrow_forwardFind the indefinite integral. Check Answer: 7x 4 + 1x dxarrow_forward
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