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3.6E: Exercises for Section 3.6

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In exercises 1 - 6, given y=f(u) and u=g(x), find dydx by using Leibniz’s notation for the chain rule: dydx=dydududx.

1) y=3u6,u=2x2

2) y=6u3,u=7x4

Answer
dydx=18u27=18(7x4)27=126(7x4)2

3) y=sinu,u=5x1

4) y=cosu,u=x8

Answer
dydx=sinu(18)=18sin(x8)

5) y=tanu,u=9x+2

6) y=4u+3,u=x26x

Answer
dydx=8x2424u+3=4x124x224x+3

For each of the following exercises,

a. decompose each function in the form y=f(u) and u=g(x), and

b. find dydx as a function of x.

7) y=(3x2)6

8) y=(3x2+1)3

Answer
a. f(u)=u3,u=3x2+1;

b. dydx=18x(3x2+1)2

9) y=sin5(x)

10) y=(x7+7x)7

Answer
a. f(u)=u7,u=x7+7x;

b. dydx=7(x7+7x)6(177x2)

11) y=tan(secx)

12) y=csc(πx+1)

Answer
a. f(u)=cscu,u=πx+1;

b. dydx=πcsc(πx+1)cot(πx+1)

13) y=cot2x

14) y=6sin3x

Answer
a. f(u)=6u3,u=sinx;

b. dydx=18sin4xcosx

In exercises 15 - 24, find dydx for each function.

15) y=(3x2+3x1)4

16) y=(52x)2

Answer
dydx=4(52x)3

17) y=cos3(πx)

18) y=(2x3x2+6x+1)3

Answer
dydx=6(2x3x2+6x+1)2(3x2x+3)

19) y=1sin2(x)

20) y=(tanx+sinx)3

Answer
dydx=3(tanx+sinx)4(sec2x+cosx)

21) y=x2cos4x

22) y=sin(cos7x)

Answer
dydx=7cos(cos7x)sin7x

23) y=6+secπx2

24) y=cot3(4x+1)

Answer
dydx=12cot2(4x+1)csc2(4x+1)

25) Let y=[f(x)]3 and suppose that f(1)=4 and dydx=10 for x=1. Find f(1).

26) Let y=(f(x)+5x2)4 and suppose that f(1)=4 and dydx=3 when x=1. Find f(1)

Answer
f(1)=1034

27) Let y=(f(u)+3x)2 and u=x32x. If f(4)=6 and dydx=18 when x=2, find f(4).

28) [T] Find the equation of the tangent line to y=sin(x2) at the origin. Use a calculator to graph the function and the tangent line together.

Answer
y=12x

29) [T] Find the equation of the tangent line to y=(3x+1x)2 at the point (1,16). Use a calculator to graph the function and the tangent line together.

30) Find the x -coordinates at which the tangent line to y=(x6x)8 is horizontal.

Answer
x=±6

31) [T] Find an equation of the line that is normal to g(θ)=sin2(πθ) at the point (14,12). Use a calculator to graph the function and the normal line together.

For exercises 32 - 39, use the information in the following table to find h(a) at the given value for a.

x f(x) f(x) g(x) g(x)
0 2 5 0 2
1 1 −2 3 0
2 4 4 1 −1
3 3 −3 2 3

32) h(x)=f(g(x));a=0

Answer
h(0)=10

33) h(x)=g(f(x));a=0

34) h(x)=(x4+g(x))2;a=1

Answer
h(1)=18

35) h(x)=(f(x)g(x))2;a=3

36) h(x)=f(x+f(x));a=1

Answer
h(1)=4

37) h(x)=(1+g(x))3;a=2

38) h(x)=g(2+f(x2));a=1

Answer
h(1)=12

39) h(x)=f(g(sinx));a=0

40) [T] The position function of a freight train is given by s(t)=100(t+1)2, with s in meters and t in seconds. At time t=6 s, find the train’s

a. velocity and

b. acceleration.

c. Considering your results in parts a. and b., is the train speeding up or slowing down?

Answer
a. v(6)=200343 m/s,

b. a(6)=6002401m/s2,

c. The train is slowing down since velocity and acceleration have opposite signs.

41) [T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and s is in inches:

s(t)=3cos(πt+π4).

a. Determine the position of the spring at t=1.5 s.

b. Find the velocity of the spring at t=1.5 s.

42) [T] The total cost to produce x boxes of Thin Mint Girl Scout cookies is C dollars, where C=0.0001x30.02x2+3x+300. In t weeks production is estimated to be x=1600+100t boxes.

a. Find the marginal cost C(x).

b. Use Leibniz’s notation for the chain rule, dCdt=dCdxdxdt, to find the rate with respect to time t that the cost is changing.

c. Use your result in part b. to determine how fast costs are increasing when t=2 weeks. Include units with the answer.

Answer
a. C(x)=0.0003x20.04x+3

b. dCdt=100(0.0003x20.04x+3)=100(0.0003(1600+100t)20.04(1600+100t)+3)=300t2+9200t+70700

c. Approximately $90,300 per week

43) [T] The formula for the area of a circle is A=πr2, where r is the radius of the circle. Suppose a circle is expanding, meaning that both the area A and the radius r (in inches) are expanding.

a. Suppose r=2100(t+7)2 where t is time in seconds. Use the chain rule dAdt=dAdrdrdt to find the rate at which the area is expanding.

b. Use your result in part a. to find the rate at which the area is expanding at t=4 s.

44) [T] The formula for the volume of a sphere is S=43πr3, where r (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

a. Suppose r=1(t+1)2112 where t is time in minutes. Use the chain rule dSdt=dSdrdrdt to find the rate at which the snowball is melting.

b. Use your result in part a. to find the rate at which the volume is changing at t=1 min.

Answer
a. dSdt=8πr2(t+1)3=8π(1(t+1)2112)2(t+1)3

b. The volume is decreasing at a rate of π36ft3/min

45) [T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function T(x)=9410cos[π12(x2)], where x is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.

46) [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function D(t)=5sin(π6t7π6)+8, where t is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

Answer
 2.3 ft/hr

3.6E: Exercises for Section 3.6 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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