1. [latex]y=3u-6, \,\,\, u=2x^2[/latex]

2. [latex]y=6u^3, \,\,\, u=7x-4[/latex]

3. [latex]y= \sin u, \,\,\, u=5x-1[/latex]

4. [latex]y= \cos u, \,\,\, u=\dfrac{−x}{8}[/latex]

5. [latex]y= \tan u, \,\,\, u=9x+2[/latex]

6. [latex]y=\sqrt{4u+3}, \,\,\, u=x^2-6x[/latex]

7. [latex]y=(3x-2)^6[/latex]

8. [latex]y=(3x^2+1)^3[/latex]

9. [latex]y= \sin^5 (x)[/latex]

10. [latex]y=\left(\dfrac{x}{7}+\dfrac{7}{x}\right)^7[/latex]

11. [latex]y= \tan ( \sec x)[/latex]

12. [latex]y= \csc (\pi x+1)[/latex]

13. [latex]y= \cot^2 x[/latex]

14. [latex]y=-6 \sin^{-3} x[/latex]

15. [latex]y=(3x^2+3x-1)^4[/latex]

16. [latex]y=(5-2x)^{-2}[/latex]

17. [latex]y= \cos^3 (\pi x)[/latex]

18. [latex]y=(2x^3-x^2+6x+1)^3[/latex]

19. [latex]y=\dfrac{1}{\sin^2 (x)}[/latex]

20. [latex]y=(\tan x+ \sin x)^{-3}[/latex]

21. [latex]y=x^2 \cos^4 x[/latex]

22. [latex]y= \sin (\cos 7x)[/latex]

23. [latex]y=\sqrt{6+ \sec \pi x^2}[/latex]

24. [latex]y= \cot^3 (4x+1)[/latex]

25. Let [latex]y=(f(x))^3[/latex] and suppose that [latex]f^{\prime}(1)=4[/latex] and [latex]\frac{dy}{dx}=10[/latex] for [latex]x=1[/latex]. Find [latex]f(1)[/latex].

26. Let [latex]y=(f(x)+5x^2)^4[/latex] and suppose that [latex]f(-1)=-4[/latex] and [latex]\frac{dy}{dx}=3[/latex] when [latex]x=-1[/latex]. Find [latex]f^{\prime}(-1)[/latex]

27. Let [latex]y=(f(u)+3x)^2[/latex] and [latex]u=x^3-2x[/latex]. If [latex]f(4)=6[/latex] and [latex]\frac{dy}{dx}=18[/latex] when [latex]x=2[/latex], find [latex]f^{\prime}(4)[/latex].

28. [T] Find the equation of the tangent line to [latex]y=−\sin \left(\dfrac{x}{2}\right)[/latex] at the origin. Use a calculator to graph the function and the tangent line together.

29. [T] Find the equation of the tangent line to [latex]y=\left(3x+\dfrac{1}{x}\right)^2[/latex] at the point [latex](1,16)[/latex]. Use a calculator to graph the function and the tangent line together.

30. Find the [latex]x[/latex]-coordinates at which the tangent line to [latex]y=\left(x-\dfrac{6}{x}\right)^8[/latex] is horizontal.

31. [T] Find an equation of the line that is normal to [latex]g(\theta)= \sin^2 (\pi \theta)[/latex] at the point [latex]\left(\frac{1}{4},\frac{1}{2}\right)[/latex]. Use a calculator to graph the function and the normal line together.

32. [latex]h(x)=f(g(x)); \,\,\, a=0[/latex]

33. [latex]h(x)=g(f(x)); \,\,\, a=0[/latex]

34. [latex]h(x)=(x^4+g(x))^{-2}; \,\,\, a=1[/latex]

35. [latex]h(x)=\left(\dfrac{f(x)}{g(x)}\right)^2; \,\,\, a=3[/latex]

36. [latex]h(x)=f(x+f(x)); \,\,\, a=1[/latex]

37. [latex]h(x)=(1+g(x))^3; \, a=2[/latex]

38. [latex]h(x)=g(2+f(x^2)); \, a=1[/latex]

39. [latex]h(x)=f(g(\sin x)); \, a=0[/latex]

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

1. velocity and
2. acceleration.
3. Using a. and b. is the train speeding up or slowing down?

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

[latex]s(t)=-3 \cos \left(\pi t+\dfrac{\pi}{4}\right)[/latex].

1. Determine the position of the spring at [latex]t=1.5[/latex] s.
2. Find the velocity of the spring at [latex]t=1.5[/latex] s.

42. [T] The total cost to produce [latex]x[/latex] boxes of Thin Mint Girl Scout cookies is [latex]C[/latex] dollars, where [latex]C=0.0001x^3-0.02x^2+3x+300[/latex]. In [latex]t[/latex] weeks production is estimated to be [latex]x=1600+100t[/latex] boxes.

1. Find the marginal cost [latex]C^{\prime}(x)[/latex].
2. Use Leibniz’s notation for the chain rule, [latex]\frac{dC}{dt}=\frac{dC}{dx} \cdot \frac{dx}{dt}[/latex], to find the rate with respect to time [latex]t[/latex] that the cost is changing.
3. Use b. to determine how fast costs are increasing when [latex]t=2[/latex] weeks. Include units with the answer.

43. [T] The formula for the area of a circle is [latex]A=\pi r^2[/latex], where [latex]r[/latex] is the radius of the circle. Suppose a circle is expanding, meaning that both the area [latex]A[/latex] and the radius [latex]r[/latex] (in inches) are expanding.

1. Suppose [latex]r=2-\dfrac{100}{(t+7)^2}[/latex] where [latex]t[/latex] is time in seconds. Use the chain rule [latex]\frac{dA}{dt}=\frac{dA}{dr} \cdot \frac{dr}{dt}[/latex] to find the rate at which the area is expanding.
2. Use a. to find the rate at which the area is expanding at [latex]t=4[/latex] s.

44. [T] The formula for the volume of a sphere is [latex]S=\frac{4}{3}\pi r^3[/latex], where [latex]r[/latex] (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

1. Suppose [latex]r=\dfrac{1}{(t+1)^2}-\dfrac{1}{12}[/latex] where [latex]t[/latex] is time in minutes. Use the chain rule [latex]\frac{dS}{dt}=\frac{dS}{dr} \cdot \frac{dr}{dt}[/latex] to find the rate at which the snowball is melting.
2. Use a. to find the rate at which the volume is changing at [latex]t=1[/latex] min.

45. [T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function [latex]T(x)=94-10 \cos \left[\dfrac{\pi}{12}(x-2)\right][/latex], where [latex]x[/latex] 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 [latex]D(t)=5 \sin \left(\dfrac{\pi}{6} t-\dfrac{7\pi}{6}\right)+8[/latex], where [latex]t[/latex] is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.