Calculus, Early Transcendentals
9th Edition
ISBN: 9781337613927
Author: Stewart
Publisher: CENGAGE L
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Chapter 3.8, Problem 20E
(a)
(i)
To determine
Find the amount due at the end of
(i)
To determine
Find the amount due at the end of
(ii)
To determine
Find the amount due at the end of
(iii)
To determine
Find the amount due at the end of
(iv)
To determine
Find the amount due at the end of
(v)
To determine
Find the amount due at the end of
(vi)
To determine
Find the amount due at the end of
(vii)
To determine
Find the amount due at the end of
(b)
To determine
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Assignment #1
Q1: Test the following series for convergence. Specify the test you use:
1
n+5
(-1)n
a) Σn=o
√n²+1
b) Σn=1 n√n+3
c) Σn=1 (2n+1)3
3n
1
d) Σn=1 3n-1
e) Σn=1
4+4n
answer problem 1a, 1b, 1c, 1d, and 1e and show work/ explain how you got the answer
Provethat
a) prove that for any irrational numbers there exists?
asequence of rational numbers Xn converg to S.
b) let S: RR be a sunctions-t.
f(x)=(x-1) arc tan (x), xe Q
3(x-1)
1+x²
x&Q
Show that lim f(x)= 0
14x
C) For any set A define the set -A=y
Chapter 3 Solutions
Calculus, Early Transcendentals
Ch. 3.1 - (a) How is the number e defined? (b) Use a...Ch. 3.1 - (a) Sketch, by hand, the graph of the function...Ch. 3.1 - Differentiate the function. 3. g(x)=4x+7Ch. 3.1 - Differentiate the function. 4. g(t)=5t+4t2Ch. 3.1 - Differentiate the function. 5. f(x)=x75x+3Ch. 3.1 - Differentiate the function. g(x)=74x23x+12Ch. 3.1 - Differentiate the function. 7. f(t)=2etCh. 3.1 - Differentiate the function. 8. F(t)=t3+e3Ch. 3.1 - Differentiate the function. 9. W()=1.83Ch. 3.1 - Differentiate the function. 10. r(z)=z5z1/2
Ch. 3.1 - Differentiate the function. 11. f(x)=x3/2+x3Ch. 3.1 - Differentiate the function. 12. V(t)=t3/5+t4Ch. 3.1 - Differentiate the function. 13. s(t)=1t+1t2Ch. 3.1 - Differentiate the function. 14. r(t)=at2+bt4Ch. 3.1 - Differentiate the function. 15. ] y=2x+xCh. 3.1 - Differentiate the function. 16. h(w)=2w2Ch. 3.1 - Differentiate the function. 17. g(x)=1x+x4Ch. 3.1 - Differentiate the function. 18. W(t)=t2etCh. 3.1 - Differentiate the function. 19. f(x)=x3(x+3)Ch. 3.1 - Differentiate the function. 20. F(t)=(2t3)2Ch. 3.1 - Differentiate the function. 21. y=3ex+4x3Ch. 3.1 - Prob. 22ECh. 3.1 - Differentiate the function. 23. f(x)=3x2+x3xCh. 3.1 - Differentiate the function. 24. y=x+xx2Ch. 3.1 - Differentiate the function. 25. G(r)=3r3/2+r5/2rCh. 3.1 - Differentiate the function. 26. G(t)=5t+7tCh. 3.1 - Differentiate the function. 27. j(x)=x2.4+e2.4Ch. 3.1 - Differentiate the function. 28. k(r)=er+reCh. 3.1 - Differentiate the function. 29. F(z)=A+Bz+Cz2z2Ch. 3.1 - Prob. 30ECh. 3.1 - Prob. 31ECh. 3.1 - Differentiate the function. 32. f()=32evCh. 3.1 - Differentiate the function. 33. P(w)=2w2w+4wCh. 3.1 - Differentiate the function. y = ex + 1 + 1Ch. 3.1 - Prob. 35ECh. 3.1 - Prob. 36ECh. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find equations of the tangent line and normal line...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find f'(x). Compare the graphs of f and f' and use...Ch. 3.1 - Find f'(x). Compare the graphs of f and f' and use...Ch. 3.1 - Prob. 47ECh. 3.1 - Prob. 48ECh. 3.1 - Find the first and second derivatives of the...Ch. 3.1 - Find the first and second derivatives of the...Ch. 3.1 - Find the first and second derivatives of the...Ch. 3.1 - Find the first and second derivatives of the...Ch. 3.1 - The equation of motion of a particle is s = t3 ...Ch. 3.1 - The equation of motion of a particle is s = t4 ...Ch. 3.1 - Biologists have proposed a cubic polynomial to...Ch. 3.1 - The number of tree species S in a given area A in...Ch. 3.1 - Boyles Law states that when a sample of gas is...Ch. 3.1 - Prob. 58ECh. 3.1 - Prob. 59ECh. 3.1 - For what value of x does the graph of f(x) = ex ...Ch. 3.1 - Show that the curve y = 2ex + 3x + 5x3 has no...Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Find equations of both lines that are tangent to...Ch. 3.1 - At what point on the curve y = 1 + 2ex 3x is the...Ch. 3.1 - Find an equation of the normal line to the curve...Ch. 3.1 - Where does the normal line to the parabola y = x2 ...Ch. 3.1 - Draw a diagram to show that there are two tangent...Ch. 3.1 - (a) Find equations of both lines through the point...Ch. 3.1 - Use the definition of a derivative to show that if...Ch. 3.1 - Find the nth derivative of each function by...Ch. 3.1 - Find a second-degree polynomial P such that P(2) =...Ch. 3.1 - The equation y" + y' 2y = x2 is called a...Ch. 3.1 - Find a cubic function y = ax3 + bx2 + cx + d whose...Ch. 3.1 - Find a parabola with equation y = ax2 + bx + c...Ch. 3.1 - Let {x2+1ifx1x+1ifx1 Is f differentiable at 1?...Ch. 3.1 - At what numbers is the following function g...Ch. 3.1 - (a) For what values of x is the function f(x) =...Ch. 3.1 - Where is the function h(x) = |x 1| + |x + 2|...Ch. 3.1 - Find the parabola with equation y = ax2 + bx whose...Ch. 3.1 - Suppose the curve y = x4 + ax3 + bx2 + cx + d has...Ch. 3.1 - For what values of a and b is the line 2x + y = b...Ch. 3.1 - Prob. 82ECh. 3.1 - What is the value of c such that the line y = 2x +...Ch. 3.1 - The graph of any quadratic function f(x) = ax2 +...Ch. 3.1 - Let f(x){x2ifx2mx+bifx2 Find the values of m and b...Ch. 3.1 - Find numbers a and b such that the given function...Ch. 3.1 - Prob. 87ECh. 3.1 - Prob. 88ECh. 3.1 - Prob. 89ECh. 3.1 - Prob. 90ECh. 3.1 - If c12, how many lines through the point (0, c)...Ch. 3.1 - Prob. 1APCh. 3.2 - Find the derivative of f(x) = (1 + 2x2)(x x2) in...Ch. 3.2 - Find the derivative o f the function...Ch. 3.2 - Differentiate. 3. y=4x2+3(2x+5)Ch. 3.2 - Differentiate. 4. y=10x2+7x22x2Ch. 3.2 - Differentiate. 5. y=x3exCh. 3.2 - Differentiate. 6. y=ex+22ex1Ch. 3.2 - Differentiate. 7. f(x)=3x25xexCh. 3.2 - Differentiate. 8. g(x)=(x+2x)exCh. 3.2 - Differentiate. 9. y=xexCh. 3.2 - Prob. 10ECh. 3.2 - Differentiate. 11. g(t)=32t5t+1Ch. 3.2 - Differentiate. 12. G(u)=6u45uu+1Ch. 3.2 - Differentiate. 13. f(t)=5tt3t1Ch. 3.2 - Prob. 14ECh. 3.2 - Differentiate. 15. y=sss2Ch. 3.2 - Differentiate. 16. y=xx+1Ch. 3.2 - Differentiate. 17. J(u)=1u+1u2u+1uCh. 3.2 - Prob. 18ECh. 3.2 - Differentiate. 19. H(u)=(uu)(u+u)Ch. 3.2 - Prob. 20ECh. 3.2 - Differentiate. 21. V(t)=t+2ettCh. 3.2 - Prob. 22ECh. 3.2 - Differentiate. 23. y=ep(p+pp)Ch. 3.2 - Differentiate. f(t)=t3t3Ch. 3.2 - Differentiate. 26. y=z2+ezzCh. 3.2 - Differentiate. f(x)=x2exx2+exCh. 3.2 - Prob. 28ECh. 3.2 - Differentiate. f(x)=xx+cxCh. 3.2 - Differentiate. f(x)=ax+bcx+dCh. 3.2 - Find f'(x) and f"(x). f(x)=xexCh. 3.2 - Find an equation of the tangent line to the given...Ch. 3.2 - (a) The curve y = 1/(1 + x2) is called a witch of...Ch. 3.2 - (a) The curve y = x/(1 + x2) is called a...Ch. 3.2 - (a) If f(x) = (x3 x)ex, find f'(x). (b) Check to...Ch. 3.2 - (a) If f(x) = (x2 1)/(x2 + 1), find f'(x) and...Ch. 3.2 - If f(x) = x2/(l + x), find f"(1).Ch. 3.2 - If g(x) = x/ex. find g(n)(x).Ch. 3.2 - Suppose that f(5) = 1, f'(5) = 6, g(5) = 3, and...Ch. 3.2 - Suppose that f(4) = 2, g(4) = 5, f'(4) = 6. and...Ch. 3.2 - If f(x) = exg(x), where g(0) = 2 and g'(0) = 5,...Ch. 3.2 - If h(2) = 4 and h'(2) = 3, find ddx(h(x)x)|x=2Ch. 3.2 - If g(x) = xf(x), where f(3) = 4 and f'(3) = 2,...Ch. 3.2 - If f(2) = 10 and f'(x) = x2f(x) for all x, find...Ch. 3.2 - If f and g are the functions whose graphs are...Ch. 3.2 - Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F...Ch. 3.2 - If g is a differentiable function, find an...Ch. 3.2 - If f is a differentiable function, find an...Ch. 3.2 - How many tangent lines to the curve y = x/(x + 1)...Ch. 3.2 - Find equations of the tangent lines to the curve...Ch. 3.2 - Find R'(0), where R(x)=x3x3+5x51+3x3+6x6+9x9 Hint:...Ch. 3.2 - Prob. 58ECh. 3.2 - A manufacturer produces bolts of a fabric with a...Ch. 3.2 - The Michaelis-Menten equation for the enzyme...Ch. 3.2 - Prob. 62ECh. 3.2 - Extended Product Rule The Product Rule can be...Ch. 3.2 - (a) If F(x) = f(x) g(x), where f and g have...Ch. 3.2 - Prob. 65ECh. 3.2 - Prob. 66ECh. 3.3 - Differentiate. 1. f(x)=3sinx2cosxCh. 3.3 - Differentiate. 2. f(x)=tanx4sinxCh. 3.3 - Differentiate. 3. y=x2+cotxCh. 3.3 - Differentiate. y = 2 sec x csc xCh. 3.3 - Prob. 5ECh. 3.3 - Differentiate. 6. g(x)=3x+x2cosxCh. 3.3 - Differentiate. y = sec tanCh. 3.3 - Differentiate. y = sin cosCh. 3.3 - Differentiate. 9. f()=(cos)sinCh. 3.3 - Differentiate. g() = e(tan )Ch. 3.3 - Differentiate. 11. H(t)=cos2tCh. 3.3 - Differentiate. 12. f(x)=exsinx+cosxCh. 3.3 - Differentiate f()=sin1+cosCh. 3.3 - Differentiate. y=cosx1sinxCh. 3.3 - Differentiate. 15. y=x2tanxCh. 3.3 - Differentiate. f(t)=cottetCh. 3.3 - Differentiate. 17. f(w)=1+secw1secwCh. 3.3 - Differentiate. y=sint1+tantCh. 3.3 - Differentiate. y=tsint1+tCh. 3.3 - Prob. 20ECh. 3.3 - Differentiate. f() = cos sinCh. 3.3 - Differentiate. f(t) = tet cot tCh. 3.3 - Show that ddx(cscx)=cscxcotx .Ch. 3.3 - Show that ddx(secx)=secxtanx .Ch. 3.3 - Show that ddx(cotx)=csc2x .Ch. 3.3 - Prove, using the definition of a derivative, that...Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Prob. 29ECh. 3.3 - (a) Find an equation of the tangent line to the...Ch. 3.3 - Prob. 32ECh. 3.3 - (a) If f(x) = sec x x, find f'(x). (b) Check to...Ch. 3.3 - (a) If f(x) = ex cos x, find f'(x) and f"(x). (b)...Ch. 3.3 - If f(t) = sec t, find f"(/4).Ch. 3.3 - (a) Use the Quotient Rule to differentiate the...Ch. 3.3 - Suppose f(/3) = 4 and f'(/3) = 2, and let g(x) =...Ch. 3.3 - For what values of x does the graph of f have a...Ch. 3.3 - For what values of x does the graph of f have a...Ch. 3.3 - A mass on a spring vibrates horizontally on a...Ch. 3.3 - An elastic band is hung on a hook and a mass is...Ch. 3.3 - A ladder 10 ft long rests against a vertical wall....Ch. 3.3 - An object with weight W is dragged along a...Ch. 3.3 - Find the limit. limx0sin5x3xCh. 3.3 - Find the limit. limx0sinxsinxCh. 3.3 - Prob. 53ECh. 3.3 - Find the limit. limx0sin3xsin5xx2Ch. 3.3 - Prob. 55ECh. 3.3 - Find the limit. limx0cscxsin(sinx)Ch. 3.3 - Prob. 57ECh. 3.3 - Find the limit. limx0sin(x2)xCh. 3.3 - Prob. 59ECh. 3.3 - Prob. 60ECh. 3.3 - Find the given derivative by finding the first few...Ch. 3.3 - Find the given derivative by finding the first few...Ch. 3.3 - Find constants A and B such that the function y =...Ch. 3.3 - (a) Evaluate limxxsin1x. (b) Evaluate limx0xsin1x....Ch. 3.3 - Differentiate each trigonometric identity to...Ch. 3.3 - A semicircle with diameter PQ sits on an isosceles...Ch. 3.3 - The figure shows a circular arc of length s and a...Ch. 3.3 - Prob. 68ECh. 3.4 - Write the composite function in the form f(g(x)) ....Ch. 3.4 - Prob. 2ECh. 3.4 - Write the composite function in the form f(g(x)) ....Ch. 3.4 - Write the composite function in the form f(g(x)) ....Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Write the composite function in the form f(g(x)) ....Ch. 3.4 - Find the derivative of the function. 7....Ch. 3.4 - Find the derivative of the function. 8....Ch. 3.4 - Find the derivative of the function. f(x)=5x+1Ch. 3.4 - Find the derivative of the function. f(x)=1x213Ch. 3.4 - Find the derivative of the function. 11....Ch. 3.4 - Prob. 12ECh. 3.4 - Find the derivative of the function. f() = cos(2)Ch. 3.4 - Find the derivative of the function. g() = cos2Ch. 3.4 - Find the derivative of the function. 15. g(x)=ex2xCh. 3.4 - Find the derivative of the function. 16. y=5xCh. 3.4 - Find the derivative of the function. y = x2e3xCh. 3.4 - Find the derivative of the function. f(t) = t sin ...Ch. 3.4 - Find the derivative of the function. f(t) = eat...Ch. 3.4 - Find the derivative of the function. 21....Ch. 3.4 - Find the derivative of the function. 22....Ch. 3.4 - Find the derivative of the function. y=xx+1Ch. 3.4 - Find the derivative of the function. y=(x+1x)5Ch. 3.4 - Find the derivative of the function. y = e tanCh. 3.4 - Find the derivative of the function. f(t)2t3Ch. 3.4 - Find the derivative of the function....Ch. 3.4 - Find the derivative of the function....Ch. 3.4 - Find the derivative of the function. r(t)=10t2Ch. 3.4 - Find the derivative of the function. f(z) =...Ch. 3.4 - Find the derivative of the function....Ch. 3.4 - Find the derivative of the function. J() = tan2(n)Ch. 3.4 - Find the derivative of the function. F(t) = et sin...Ch. 3.4 - Find the derivative of the function. F(t)=t2t3+1Ch. 3.4 - Find the derivative of the function. G(x) = 4C/xCh. 3.4 - Prob. 36ECh. 3.4 - Prob. 37ECh. 3.4 - Prob. 38ECh. 3.4 - Find the derivative of the function. 39....Ch. 3.4 - Prob. 40ECh. 3.4 - Find the derivative of the function. 41. y= sin 2...Ch. 3.4 - Find the derivative of the function. y = esin 2x +...Ch. 3.4 - Find the derivative of the function. 43....Ch. 3.4 - Find the derivative of the function. 44....Ch. 3.4 - Find the derivative of the function. f(t) =...Ch. 3.4 - Find the derivative of the function. y=x+x+xCh. 3.4 - Find the derivative of the function. 47....Ch. 3.4 - Find the derivative of the function. y=234xCh. 3.4 - Find the derivative of the function. 49....Ch. 3.4 - Find the derivative of the function. 50....Ch. 3.4 - Find the derivative of the function....Ch. 3.4 - Find the derivative of the function. 52....Ch. 3.4 - Find y and y. y = cos(sin 3)Ch. 3.4 - Find y and y . 54. y=(1+x)3Ch. 3.4 - Find y and y. y=eexCh. 3.4 - Find an equation of the tangent line to the curve...Ch. 3.4 - Find an equation of the tangent line to the curve...Ch. 3.4 - Find an equation of the tangent line to the curve...Ch. 3.4 - Find an equation of the tangent line to the curve...Ch. 3.4 - Prob. 61ECh. 3.4 - (a) The curve y=|x|/2x2 is called a bullet-nose...Ch. 3.4 - (a) If f(x)=2x2x, find f(x). (b) Check to see that...Ch. 3.4 - Prob. 64ECh. 3.4 - Find all points on the graph of the function f(x)...Ch. 3.4 - At what point on the curve y=1+2x is the tangent...Ch. 3.4 - If F(x) = f(g(x)), where f(2) = 8, f(2) = 4, f(5)...Ch. 3.4 - If h(x)=4+3f(x), where f(1) = 7andf(1) = 4, find...Ch. 3.4 - A table of values for f, g, f, and g is given. (a)...Ch. 3.4 - Let f and g be the functions in Exercise 69. (a)...Ch. 3.4 - If f and g are the functions whose graphs are...Ch. 3.4 - If f is the function whose graph is shown, let...Ch. 3.4 - If g(x)=f(x), where the graph off is shown,...Ch. 3.4 - Prob. 74ECh. 3.4 - Suppose f is differentiable on . Let F(x) = f(ex)...Ch. 3.4 - Let g(x) = ecx + f(x) and h(x) = ekxf(x), where...Ch. 3.4 - Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 3,...Ch. 3.4 - If g is a twice differentiable function and f(x) =...Ch. 3.4 - Prob. 79ECh. 3.4 - Prob. 80ECh. 3.4 - Show that the function y = e2x (A cos 3x + B sin...Ch. 3.4 - For what values of r does the function y = erx...Ch. 3.4 - Find the 50th derivative of y = cos 2x.Ch. 3.4 - Prob. 84ECh. 3.4 - The displacement of a particle on a vibrating...Ch. 3.4 - If the equation of motion of a particle is given...Ch. 3.4 - A Cepheid variable star is a star whose brightness...Ch. 3.4 - In Example 1.3.4 we arrived at a model for the...Ch. 3.4 - Prob. 89ECh. 3.4 - Under certain circumstance a rumor spreads...Ch. 3.4 - The average blood alcohol concentration (BAC) of...Ch. 3.4 - Air is being pumped into a spherical weather...Ch. 3.4 - Prob. 93ECh. 3.4 - Prob. 94ECh. 3.4 - Use the Chain Rule to prove the following. (a) The...Ch. 3.4 - Prob. 96ECh. 3.4 - Use the Chain Rule to show that if is measured in...Ch. 3.4 - Prob. 98ECh. 3.4 - Prob. 99ECh. 3.4 - Prob. 101ECh. 3.4 - Prob. 102ECh. 3.4 - An approach path for an aircraft landing is shown...Ch. 3.4 - Prob. 2APCh. 3.4 - Prob. 3APCh. 3.5 - (a) Find y by implicit differentiation. (b) Solve...Ch. 3.5 - (a) Find y by implicit differentiation. (b) Solve...Ch. 3.5 - (a) Find y by implicit differentiation. (b) Solve...Ch. 3.5 - (a) Find y by implicit differentiation. (b) Solve...Ch. 3.5 - Find dy/dx by implicit differentiation. 5. x2 4xy...Ch. 3.5 - Find dy/dx by implicit differentiation. 6. 2x2 +...Ch. 3.5 - Find dy/dx by implicit differentiation. 7. x4 +...Ch. 3.5 - Find dy/dx by implicit differentiation. 8. x3 xy2...Ch. 3.5 - Find dy/dx by implicit differentiation. 9....Ch. 3.5 - Find dy/dx by implicit differentiation. 10. xey =...Ch. 3.5 - Find dy/dx by implicit differentiation. 11. y cos...Ch. 3.5 - Find dy/dx by implicit differentiation. 13....Ch. 3.5 - Find dy/dx by implicit differentiation. 16....Ch. 3.5 - Find dy/dx by implicit differentiation. 15. ex/y...Ch. 3.5 - If f(x) + x2 [f(x)]3 = 10 and f(1) = 2, find f(1).Ch. 3.5 - If g(x) + x sin g(x) = x2, find g(0).Ch. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 28ECh. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 34ECh. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - (a) The curve with equation y2 = 5x4 x2 is called...Ch. 3.5 - (a) The curve with equation y2 = x3 + 3x2 is...Ch. 3.5 - Find y by implicit differentiation. 35. x2 + 4y2 =...Ch. 3.5 - Find y by implicit differentiation. 36. x2 + xy +...Ch. 3.5 - Find y by implicit differentiation. 37. sin y +...Ch. 3.5 - Find y by implicit differentiation. 38. x3 y3 = 7Ch. 3.5 - If xy + ey = e, find the value of y at the point...Ch. 3.5 - If x2 + xy + y3 = 1, find the value of y at the...Ch. 3.5 - Prob. 45ECh. 3.5 - (a) The curve with equation 2y3+y2y5=x42x3+x2 has...Ch. 3.5 - Find the points on the lemniscate in Exercise 31...Ch. 3.5 - Show by implicit differentiation that the tangent...Ch. 3.5 - Find an equation of the tangent line to the...Ch. 3.5 - Show that the sum of the x-and y-intercepts of any...Ch. 3.5 - Show, using implicit differentiation, that any...Ch. 3.5 - The Power Rule can be proved using implicit...Ch. 3.5 - Prob. 53ECh. 3.5 - Orthogonal Trajectories Two curves are orthogonal...Ch. 3.5 - Prob. 55ECh. 3.5 - Orthogonal Trajectories Two curves are orthogonal...Ch. 3.5 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 3.5 - Prob. 58ECh. 3.5 - (a) The van der Waals equation for n moles of a...Ch. 3.5 - The equation x2 xy + y2 = 3 re presents a...Ch. 3.5 - (a) Where does the normal line to the ellipse x2 ...Ch. 3.5 - Find all points on the curve x2y2 + xy = 2 where...Ch. 3.5 - Find equations of both the tangent lines to the...Ch. 3.5 - Use implicit differentiation to find dy/dx for the...Ch. 3.5 - Prob. 66ECh. 3.5 - The figure shows a lamp located three units to the...Ch. 3.5 - Prob. 1DPCh. 3.6 - Explain why the natural logarithmic function y =...Ch. 3.6 - Prob. 2ECh. 3.6 - Differentiate the function. 3. f(x)=lnx2+3x+5Ch. 3.6 - Differentiate the function. 4. f(x)=xlnxxCh. 3.6 - Differentiate the function. f(x ) = sin(ln x)Ch. 3.6 - Differentiate the function. f(x) = ln(sin2x)Ch. 3.6 - Differentiate the function. f(x)=ln1xCh. 3.6 - Differentiate the function. y=1lnxCh. 3.6 - Differentiate the function. g(x) = ln(xe2x)Ch. 3.6 - Differentiate the function. g(t)=1+lntCh. 3.6 - Differentiate the function. F(t) =(ln t)2 sin tCh. 3.6 - Differentiate the function. 12. p(t)=lnt2+1Ch. 3.6 - Differentiate the function. 13. y=log8x2+3xCh. 3.6 - Prob. 14ECh. 3.6 - Differentiate the function. F(s) = ln ln sCh. 3.6 - Differentiate the function. p(v)=lnv1vCh. 3.6 - Differentiate the function. T(z) = 2z log2zCh. 3.6 - Differentiate the function. 18. g(t)=lntt2+142t13Ch. 3.6 - Differentiate the function. y = ln(csc x cot x)Ch. 3.6 - Differentiate the function. 21. y=lnex+xexCh. 3.6 - Prob. 22ECh. 3.6 - Differentiate the function. 23. h(x)=ex2+lnxCh. 3.6 - Differentiate the function. 24. y=ln1+2x12xCh. 3.6 - Differentiate the function. y = log2 (x log5 x)Ch. 3.6 - Show that ddxlnx+x2+1=1x2+1 .Ch. 3.6 - Show that ddxln1cosx1+cosx=cscx .Ch. 3.6 - Find y and y. y=xlnxCh. 3.6 - Find y and y. y=lnx1+lnxCh. 3.6 - Find y and y. y = ln |sec x|Ch. 3.6 - Find y and y. y = ln(l + ln x)Ch. 3.6 - Differentiate f and find the domain of f....Ch. 3.6 - Differentiate f and find the domain of f....Ch. 3.6 - Differentiate f and find the domain of f. f(x) =...Ch. 3.6 - Differentiate f and find the domain of f. f(x) ln...Ch. 3.6 - If f(x) = ln(x + ln x), find f(1).Ch. 3.6 - If f(x) = cos(ln x2), find f(1).Ch. 3.6 - Find an equation of the tangent line to the curve...Ch. 3.6 - Find an equation of the tangent line to the curve...Ch. 3.6 - If f(x) = sin x + ln x, find f(x). Check that your...Ch. 3.6 - Find equations of the tangent lines to the curve y...Ch. 3.6 - Let f(x) = cx + ln(cos x). For what value of c is...Ch. 3.6 - Let f(x) = logb (3x2 2). For what value of b is...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Find y if y = ln(x2 + y2).Ch. 3.6 - Find y if xy = yx.Ch. 3.6 - Find a formula for f(n)(x) if f(x) = ln(x 1).Ch. 3.6 - Find d9dx9(x8lnx).Ch. 3.6 - Use the definition of derivative to prove that...Ch. 3.6 - Show that limn(1+xn)n=exfor any x 0.Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Prob. 73ECh. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Prob. 76ECh. 3.6 - Find the derivative of the function. Simplify...Ch. 3.6 - Prob. 78ECh. 3.6 - Prob. 79ECh. 3.6 - Prob. 80ECh. 3.6 - Prob. 81ECh. 3.6 - (a) One way of defining sec1x is to say that...Ch. 3.6 - Use the formula in Exercise 83. 84. If f(4)=5 and...Ch. 3.6 - Use the formula in Exercise 83. 84. If f(4)=5 and...Ch. 3.6 - Use the formula in Exercise 83. 85. If f(x)=x+ex ,...Ch. 3.6 - Prob. 86ECh. 3.6 - Prob. 87ECh. 3.6 - Prob. 88ECh. 3.7 - A particle moves according to a law of motion...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - Graphs of the velocity functions of two particles...Ch. 3.7 - Prob. 6ECh. 3.7 - Prob. 7ECh. 3.7 - The height (in meters) of a projectile shot...Ch. 3.7 - If a ball is thrown vertically upward with a...Ch. 3.7 - If a rock is thrown vertically upward from the...Ch. 3.7 - A particle moves with position function s = t4 ...Ch. 3.7 - (a) A company makes computer chips from square...Ch. 3.7 - (a) Sodium chlorate crystals are easy to grow in...Ch. 3.7 - (a) Find the average rate of change of the area of...Ch. 3.7 - A stone is dropped into a lake, creating a...Ch. 3.7 - A spherical balloon is being inflated. Find the...Ch. 3.7 - (a) The volume of a growing spherical cell is...Ch. 3.7 - The mass of the part of a metal rod that lies...Ch. 3.7 - The quantity of charge Q in coulombs (C) that has...Ch. 3.7 - Newtons Law of Gravitation says that the magnitude...Ch. 3.7 - Prob. 23ECh. 3.7 - Some of the highest tides in the world occur in...Ch. 3.7 - Boyles Law states that when a sample of gas is...Ch. 3.7 - If, in Example 4, one molecule of the product C is...Ch. 3.7 - In Example 6 we considered a bacteria population...Ch. 3.7 - The number of yeast cells in a laboratory culture...Ch. 3.7 - Prob. 29ECh. 3.7 - Prob. 30ECh. 3.7 - Prob. 31ECh. 3.7 - Prob. 32ECh. 3.7 - Prob. 33ECh. 3.7 - The cost function for a certain commodity is C(q)...Ch. 3.7 - Prob. 35ECh. 3.7 - If R denotes the reaction of the body to some...Ch. 3.7 - Patients undergo dialysis treatment to remove urea...Ch. 3.7 - Invasive species often display a wave of advance...Ch. 3.7 - The gas law for an ideal gas at absolute...Ch. 3.7 - In a fish farm, a population of fish is introduced...Ch. 3.7 - In the study of ecosystems, predator-prey models...Ch. 3.7 - Prob. 43ECh. 3.8 - A population of the yeast cell Saccharomyces...Ch. 3.8 - A common inhabitant of human intestines is the...Ch. 3.8 - A culture of the bacterium Salmonella enteritidis...Ch. 3.8 - A bacteria culture grows with constant relative...Ch. 3.8 - The table gives estimates of the world population,...Ch. 3.8 - Experiments show that if the chemical reaction...Ch. 3.8 - Strontium-90 has a half-life of 28 days. (a) A...Ch. 3.8 - The half-life of cesium-137 is 30 years. Suppose...Ch. 3.8 - A sample oflritium-3 decayed to 94.5% of its...Ch. 3.8 - Prob. 11ECh. 3.8 - Radiocarbon Dating Scientists can determine the...Ch. 3.8 - Prob. 13ECh. 3.8 - A curve passes through the point (0, 5) and has...Ch. 3.8 - A roast turkey is taken from an oven when its...Ch. 3.8 - In a murder investigation, the temperature of the...Ch. 3.8 - When a cold drink is taken from a refrigerator,...Ch. 3.8 - A freshly brewed cup of coffee has temperature 95C...Ch. 3.8 - Prob. 19ECh. 3.8 - Prob. 20ECh. 3.8 - Prob. 21ECh. 3.8 - Prob. 22ECh. 3.9 - (a) If V is the volume of a cube with edge length...Ch. 3.9 - (a) If A is the area of a circle with radius r and...Ch. 3.9 - Each side of a square is increasing at a rate of 6...Ch. 3.9 - The radius of a sphere is increasing at a rate of...Ch. 3.9 - The radius of a spherical ball is increasing at a...Ch. 3.9 - The length of a rectangle is increasing at a rate...Ch. 3.9 - A cylindrical tank with radius 5m is being filled...Ch. 3.9 - The area of a triangle with sides of lengths a and...Ch. 3.9 - Prob. 9ECh. 3.9 - If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4,...Ch. 3.9 - The weight w of an astronaut (in pounds) is...Ch. 3.9 - A particle is moving along a hyperbola xy = 8. As...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - (a) What quantities are given in the problem? (b)...Ch. 3.9 - Two cars start moving from the same point. One...Ch. 3.9 - A spotlight on the ground shines on a wall 12m...Ch. 3.9 - A man starts walking north at 4 ft/s from a point...Ch. 3.9 - A baseball diamond is a square with side 90 ft. A...Ch. 3.9 - The altitude of a triangle is increasing at a rate...Ch. 3.9 - A boat is pulled into a dock by a rope attached to...Ch. 3.9 - Use the fact that the distance (in meters) a...Ch. 3.9 - Prob. 24ECh. 3.9 - Water is leaking out of an inverted conical tank...Ch. 3.9 - Prob. 26ECh. 3.9 - A water trough is 10m long and a cross-section has...Ch. 3.9 - A trough is 10 ft long and its ends have the shape...Ch. 3.9 - Gravel is being dumped from a conveyor belt at a...Ch. 3.9 - A swimming pool is 20 ft wide, 40 ft long, 3 ft...Ch. 3.9 - The sides of an equilateral triangle are...Ch. 3.9 - A kite 100ft above the ground moves horizontally...Ch. 3.9 - A car is traveling north on a straight road at...Ch. 3.9 - If the minute hand of a clock has length r (in...Ch. 3.9 - How fast is the angle between the ladder and the...Ch. 3.9 - According to the model we used to solve Example 2,...Ch. 3.9 - Boyles Law states that when a sample of gas is...Ch. 3.9 - A faucet is filling a hemispherical basin of...Ch. 3.9 - If two resistors with resistances R1 and R2 are...Ch. 3.9 - When air expands adiabatically (without gaining or...Ch. 3.9 - Prob. 41ECh. 3.9 - Brain weight B as a function of body weight Win...Ch. 3.9 - Two sides of a triangle have lengths 12m and 15m ....Ch. 3.9 - Two carts, A and B, are connected by a rope 39 ft...Ch. 3.9 - A television camera is positioned 4000 ft from the...Ch. 3.9 - A lighthouse is located on a small island 3 km...Ch. 3.9 - A plane flies horizontally at an altitude of 5 km...Ch. 3.9 - A Ferris wheel with a radius of 10m is rotating at...Ch. 3.9 - A plane flying with a constant speed of 300 km/h...Ch. 3.9 - Two people start from the same point. One walks...Ch. 3.9 - A runner sprints around a circular track of radius...Ch. 3.9 - The minute hand on a watch is 8 mm long and the...Ch. 3.9 - Prob. 53ECh. 3.10 - Find the linearization L(x) of the function at n....Ch. 3.10 - Find the linearization L(x) of the function at a ....Ch. 3.10 - Find the linearization L(x) of the function at a ....Ch. 3.10 - Prob. 4ECh. 3.10 - Find the linear approximation of the function...Ch. 3.10 - Find the linear approximation of the function...Ch. 3.10 - Prob. 7ECh. 3.10 - Prob. 8ECh. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Find the differential of the function. 11. y=e5xCh. 3.10 - Find the differential of the function. 12. y=1t4Ch. 3.10 - Find the differential of the function. 13....Ch. 3.10 - Prob. 14ECh. 3.10 - Find the differential of the function. 15. y=1x23xCh. 3.10 - Prob. 16ECh. 3.10 - Find the differential of the function. 17....Ch. 3.10 - Prob. 18ECh. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - Prob. 22ECh. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Prob. 26ECh. 3.10 - Compare the values of y and dy if x changes from 1...Ch. 3.10 - Prob. 28ECh. 3.10 - Prob. 29ECh. 3.10 - Prob. 30ECh. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Explain, in terms of linear approximations or...Ch. 3.10 - Explain, in terms of linear approximations or...Ch. 3.10 - Prob. 40ECh. 3.10 - The edge of a cube was found to be 30 cm with a...Ch. 3.10 - The radius of a circular disk is given as 24 cm...Ch. 3.10 - The circumference of a sphere was measured to be...Ch. 3.10 - Use differentials to estimate the amount of paint...Ch. 3.10 - (a) Use differentials to find a formula for the...Ch. 3.10 - One side of a right triangle is known to be 20 cm...Ch. 3.10 - If a current I passes through a resistor with...Ch. 3.10 - When blood flows along a blood vessel, the flux F...Ch. 3.10 - Establish the following rules for working with...Ch. 3.10 - Suppose that the only information we have about a...Ch. 3.10 - Suppose that we dont have a formula for g(x) but...Ch. 3.10 - Prob. 1DPCh. 3.10 - Prob. 2DPCh. 3.10 - Prob. 3DPCh. 3.10 - Prob. 4DPCh. 3.10 - Prob. 5DPCh. 3.10 - Prob. 6DPCh. 3.11 - Find the numerical value of each expression. 1....Ch. 3.11 - Find the numerical value of each expression. 2....Ch. 3.11 - Find the numerical value of each expression. 3....Ch. 3.11 - Find the numerical value of each expression. 4....Ch. 3.11 - Find the numerical value of each expression. 5....Ch. 3.11 - Prob. 6ECh. 3.11 - Write 8sinhx+5coshx in terms of exandex .Ch. 3.11 - Write 2e2x+3e2x in terms of sinh2x and cosh2x .Ch. 3.11 - Write sinh(lnx) as a rational function of x .Ch. 3.11 - Prob. 10ECh. 3.11 - Prove the identity. 7. sinh(x) = sinh x (This...Ch. 3.11 - Prob. 12ECh. 3.11 - Prove the identity. 9. cosh x + sinh x = exCh. 3.11 - Prove the identity. 10. cosh x sinh r = exCh. 3.11 - Prove the identity. 11. sinh(x + y) = sinh x cosh...Ch. 3.11 - Prob. 16ECh. 3.11 - Prove the identity. 13. coth2x 1 = csch2xCh. 3.11 - Prove the identity. 14....Ch. 3.11 - Prove the identity. 15. sinh 2x = 2 sinh x cosh xCh. 3.11 - Prob. 20ECh. 3.11 - Prove the identity. 17. tanh(lnx)=x21x2+1Ch. 3.11 - Prove the identity. 18. 1+tanhx1tanhx=e2xCh. 3.11 - Prove the identity. 19. (cosh x + sinh x)n = cosh...Ch. 3.11 - If x=1213 find the values of the other hyperbolic...Ch. 3.11 - If cosh=53 and x 0. find the values of the other...Ch. 3.11 - Prob. 26ECh. 3.11 - Use the definitions of the hyperbolic functions to...Ch. 3.11 - Prove the formulas given in Table 1 for the...Ch. 3.11 - Prob. 29ECh. 3.11 - Prob. 30ECh. 3.11 - Prob. 32ECh. 3.11 - Prob. 33ECh. 3.11 - Prob. 34ECh. 3.11 - Prob. 35ECh. 3.11 - Find the derivative. Simplify where possible. 30....Ch. 3.11 - Find the derivative. Simplify where possible. 33....Ch. 3.11 - Find the derivative. Simplify where possible. 32....Ch. 3.11 - Find the derivative. Simplify where possible. 35....Ch. 3.11 - Prob. 40ECh. 3.11 - Find the derivative. Simplify where possible. 31....Ch. 3.11 - Prob. 42ECh. 3.11 - Find the derivative. Simplify where possible. 39....Ch. 3.11 - Prob. 46ECh. 3.11 - Prob. 47ECh. 3.11 - Prob. 48ECh. 3.11 - Prob. 49ECh. 3.11 - Find the derivative. Simplify where possible. 51....Ch. 3.11 - Find the derivative. Simplify where possible. 42....Ch. 3.11 - Find the derivative. Simplify where possible. 43....Ch. 3.11 - Prob. 54ECh. 3.11 - Prob. 55ECh. 3.11 - The Gateway Arch The Gateway Arch in St. Louis was...Ch. 3.11 - Prob. 57ECh. 3.11 - Prob. 58ECh. 3.11 - Prob. 59ECh. 3.11 - Prob. 60ECh. 3.11 - Prob. 61ECh. 3.11 - A model for the velocity of a falling object after...Ch. 3.11 - (a) Show that any function of the form y = A sinh...Ch. 3.11 - If x = ln( sec + tan ), show that sec = cosh x.Ch. 3.11 - At what point of the curve y = cosh x does the...Ch. 3.11 - Investigate the family of functions fn(x) = tanh...Ch. 3.11 - Prob. 67ECh. 3 - State each differentiation rule both in symbols...Ch. 3 - State the derivative of each function. (a) y = xn...Ch. 3 - (a) How is the number e defined? (b) Express e as...Ch. 3 - (a) Explain how implicit differentiation works....Ch. 3 - Give several examples of how the derivative can be...Ch. 3 - Prob. 6CCCh. 3 - (a) Write an expression for the linearization of f...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 10TFQCh. 3 - Prob. 11TFQCh. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 15TFQCh. 3 - Calculate y'. 1. y = (x2 + x3)4Ch. 3 - Calculate y'. 2. y=1x1x35Ch. 3 - Prob. 3ECh. 3 - Calculate y'. 4. y=tanx1+cosxCh. 3 - Prob. 5ECh. 3 - Prob. 6ECh. 3 - Prob. 7ECh. 3 - Calculate y'. 8. xey = y sin xCh. 3 - Calculate y'. 9. y = ln(x ln x)Ch. 3 - Prob. 10ECh. 3 - Prob. 11ECh. 3 - Calculate y'. 12. y = (arcsin 2x)2Ch. 3 - Prob. 13ECh. 3 - Calculate y'. 14. y = ln sec xCh. 3 - Prob. 15ECh. 3 - Prob. 16ECh. 3 - Calculate y'. 17. y=arctanCh. 3 - Prob. 18ECh. 3 - Prob. 19ECh. 3 - Prob. 20ECh. 3 - Calculate y'. 21. y = 3x ln xCh. 3 - Prob. 22ECh. 3 - Prob. 23ECh. 3 - Calculate y'. 24. y=1/x+x3Ch. 3 - Prob. 25ECh. 3 - Prob. 26ECh. 3 - Prob. 27ECh. 3 - Prob. 28ECh. 3 - Calculate y'. 29. y=lnsinx12sin2xCh. 3 - Prob. 30ECh. 3 - Prob. 31ECh. 3 - Prob. 32ECh. 3 - Calculate y'. 33. y = ln | sec 5x + tan 5x |Ch. 3 - Prob. 34ECh. 3 - Calculate y'. 35. y = cot(3x2 + 5)Ch. 3 - Prob. 36ECh. 3 - Prob. 37ECh. 3 - Calculate y . 38. y=xsec1xCh. 3 - Prob. 39ECh. 3 - Prob. 40ECh. 3 - Prob. 42ECh. 3 - Prob. 43ECh. 3 - Prob. 45ECh. 3 - Prob. 46ECh. 3 - Prob. 47ECh. 3 - Prob. 48ECh. 3 - Calculate y'. 45. y = ln( cosh 3x)Ch. 3 - Prob. 50ECh. 3 - Prob. 51ECh. 3 - Prob. 52ECh. 3 - Prob. 53ECh. 3 - Prob. 54ECh. 3 - Prob. 55ECh. 3 - Prob. 56ECh. 3 - Find y if x6 + y6 = 1.Ch. 3 - Find f(n)(x) if f(x) = 1/(2 x).Ch. 3 - Prob. 59ECh. 3 - Prob. 60ECh. 3 - Find an equation of the tangent to the curve at...Ch. 3 - Prob. 62ECh. 3 - Prob. 63ECh. 3 - Find equations of the tangent line and normal line...Ch. 3 - Find equations of the tangent line and normal line...Ch. 3 - If f(x) = xesin x find f(x). Graph f and f on the...Ch. 3 - Prob. 67ECh. 3 - (a) If f(x) = 4x tan x, /2 x /2, find f and f....Ch. 3 - At what points on the curve y = sin x + cos x, 0 ...Ch. 3 - Prob. 70ECh. 3 - Prob. 71ECh. 3 - Prob. 72ECh. 3 - Suppose that f(1) = 2 f(1) = 3 f(2) = 1 f'(2) = 2...Ch. 3 - If f and g are the functions whose graphs are...Ch. 3 - Find f in terms of g. f(x) = x2g(x)Ch. 3 - Prob. 76ECh. 3 - Find f in terms of g. f(x) = [g(x)]2Ch. 3 - Prob. 78ECh. 3 - Find f in terms of g. f(x) = g(ex)Ch. 3 - Prob. 80ECh. 3 - Prob. 81ECh. 3 - Find f in terms of g. f(x) = g(ln x)Ch. 3 - Prob. 83ECh. 3 - Prob. 84ECh. 3 - Find f in terms of f and g. h(x) = f(g(sin 4x))Ch. 3 - Prob. 86ECh. 3 - At what point on the curve y = [ln(x + 4)]2 is the...Ch. 3 - (a) Find an equation of the tangent to the curve y...Ch. 3 - Prob. 89ECh. 3 - The function C(t) = K(eat ebt), where a, b, and K...Ch. 3 - Prob. 91ECh. 3 - Prob. 92ECh. 3 - A particle moves on a vertical line so that its...Ch. 3 - Prob. 94ECh. 3 - Prob. 95ECh. 3 - Prob. 96ECh. 3 - A bacteria culture contains 200 cells initially...Ch. 3 - Cobalt-60 has a half-life of 5.24 years. (a) Find...Ch. 3 - Let C(t) be the concentration of a drug in the...Ch. 3 - A cup of hot chocolate has temperature 80C in a...Ch. 3 - Prob. 101ECh. 3 - A paper cup has the shape of a cone with height 10...Ch. 3 - A balloon is rising at a constant speed of 5 ft/s....Ch. 3 - Prob. 104ECh. 3 - Prob. 105ECh. 3 - (a) Find the linear approximation to f(x)=25x2...Ch. 3 - (a) Find the linearization of f(x)1+3x3 at a = 0....Ch. 3 - Prob. 108ECh. 3 - A window has the shape of a square surmounted by a...Ch. 3 - Prob. 110ECh. 3 - Prob. 111ECh. 3 - Express the limit as a derivative and evaluate....Ch. 3 - Prob. 113ECh. 3 - Suppose f is a differentiable function such that...Ch. 3 - Prob. 115ECh. 3 - Show that the length of the portion of any tangent...Ch. 3 - Find points P and Q on the parabola y = 1 x2 so...Ch. 3 - Prob. 2PCh. 3 - Prob. 3PCh. 3 - Prob. 4PCh. 3 - If f(x)=limtxsectsecxtx, find the value of f'(/4).Ch. 3 - Find the values of the constants a and b such that...Ch. 3 - Show that sin-1(tanh x) = tan1(sinh x).Ch. 3 - Prob. 8PCh. 3 - Prove that dndxn(sin4x+cos4x)=4n1cos(4x+n/2).Ch. 3 - Prob. 10PCh. 3 - Prob. 11PCh. 3 - Find all values of r such that the parabolas y =...Ch. 3 - How many lines are tangent to both of the circles...Ch. 3 - Prob. 14PCh. 3 - The figure shows a rotating wheel with radius 40...Ch. 3 - Tangent lines T1, and T2, are drawn at two points...Ch. 3 - Prob. 17PCh. 3 - Prob. 18PCh. 3 - Prob. 19PCh. 3 - Prob. 20PCh. 3 - (a) Use the identity for tan(x y) (see Equation...Ch. 3 - Prob. 22PCh. 3 - Prob. 23PCh. 3 - Prob. 24PCh. 3 - Prob. 25PCh. 3 - Prob. 26PCh. 3 - For what value of k does the equation e2x=kx have...Ch. 3 - Prob. 28PCh. 3 - If y=xa212a21arctansinxa+a21+cosx show that...Ch. 3 - Given an ellipse x2/a2 + y2/b2 = 1, where a b,...Ch. 3 - Find the two points on the curve y = x4 2x2 x...Ch. 3 - Suppose that three points on the parabola y = x2...Ch. 3 - Prob. 33PCh. 3 - Prob. 34PCh. 3 - Prob. 35P
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- Q2: Find the interval and radius of convergence for the following series: Σ n=1 (-1)η-1 xn narrow_forward8. Evaluate arctan x dx a) xartanx 2 2 In(1 + x²) + C b) xartanx + 1½-3ln(1 + x²) + C c) xartanx + In(1 + x²) + C d) (arctanx)² + C 2 9) Evaluate Inx³ dx 3 a) +C b) ln x² + C c)¾½ (lnx)² d) 3x(lnx − 1) + C - x 10) Determine which integral is obtained when the substitution x = So¹² √1 - x²dx sine is made in the integral πT π π a) √ sin cos e de b) √ cos² de c) c Ꮎ Ꮎ cos² 0 de c) cos e de d) for cos² e de πT 11. Evaluate tan³xdx 1 a) b) c) [1 - In 2] 2 2 c) [1 − In2] d)½½[1+ In 2]arrow_forward12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward
- 4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forward
- Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forward3. Consider the initial value problem 9y" +12y' + 4y = 0, y(0) = a>0: y′(0) = −1. Solve the problem and find the value of a such that the solution of the initial value problem is always positive.arrow_forward
- 5. Euler's equation. Determine the values of a for which all solutions of the equation 5 x²y" + axy' + y = 0 that have the form (A + B log x) x* or Ax¹¹ + Bä” tend to zero as a approaches 0.arrow_forward4. Problem on variable change. The purpose of this problem is to perform an appropriate change of variables in order to reduce the problem to a second-order equation with constant coefficients. ty" + (t² − 1)y'′ + t³y = 0, 0arrow_forward4. Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by f(t)== 90t 951-90 Find the rate at which the number of facts remembered is changing after 1 hour and after 10 hours. Interpret.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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