For the following exercises (1-10), find [latex]\frac{dy}{dx}[/latex] for the given functions.
1. [latex]y=x^2- \sec x+1[/latex]
2. [latex]y=3 \csc x+\dfrac{5}{x}[/latex]
3. [latex]y=x^2 \cot x[/latex]
4. [latex]y=x-x^3 \sin x[/latex]
5. [latex]y=\dfrac{\sec x}{x}[/latex]
6. [latex]y= \sin x \tan x[/latex]
7. [latex]y=(x+ \cos x)(1- \sin x)[/latex]
8. [latex]y=\dfrac{\tan x}{1- \sec x}[/latex]
9. [latex]y=\dfrac{1- \cot x}{1+ \cot x}[/latex]
10. [latex]y= \cos x(1+ \csc x)[/latex]
For the following exercises (11-16), find the equation of the tangent line to each of the given functions at the indicated values of [latex]x[/latex]. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.
11. [T] [latex]f(x)=−\sin x, \,\,\, x=0[/latex]
12. [T] [latex]f(x)= \csc x, \,\,\, x=\frac{\pi}{2}[/latex]
13. [T] [latex]f(x)=1+ \cos x, \,\,\, x=\frac{3\pi}{2}[/latex]
14. [T] [latex]f(x)= \sec x, \,\,\, x=\frac{\pi}{4}[/latex]
15. [T] [latex]f(x)=x^2- \tan x, \,\,\, x=0[/latex]
16. [T] [latex]f(x)=5 \cot x, \,\,\, x=\frac{\pi}{4}[/latex]
For the following exercises (17-22), find [latex]\frac{d^2 y}{dx^2}[/latex] for the given functions.
17. [latex]y=x \sin x- \cos x[/latex]
18. [latex]y= \sin x \cos x[/latex]
19. [latex]y=x-\frac{1}{2} \sin x[/latex]
20. [latex]y=\frac{1}{x}+ \tan x[/latex]
21. [latex]y=2 \csc x[/latex]
22. [latex]y=\sec^2 x[/latex]
23. Find all [latex]x[/latex] values on the graph of [latex]f(x)=-3 \sin x \cos x[/latex] where the tangent line is horizontal.
24. Find all [latex]x[/latex] values on the graph of [latex]f(x)=x-2 \cos x[/latex] for [latex]0<x<2\pi[/latex] where the tangent line has a slope of 2.
25. Let [latex]f(x)= \cot x[/latex]. Determine the point(s) on the graph of [latex]f[/latex] for [latex]0<x<2\pi[/latex] where the tangent line is parallel to the line [latex]y=-2x[/latex].
26. [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 \cos t[/latex] where [latex]s[/latex] is measured in inches and [latex]t[/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[/latex] s.
27. Let the position of a swinging pendulum in simple harmonic motion be given by [latex]s(t)=a \cos t+b \sin t[/latex] where [latex]a[/latex] and [latex]b[/latex] are constants, [latex]t[/latex] measures time in seconds, and [latex]s[/latex] measures position in centimeters, If the position is [latex]0[/latex]cm and the velocity is [latex]3[/latex]cm/s when [latex]t=0[/latex], find the values of [latex]a[/latex] and [latex]b[/latex].
28. After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 \cos t[/latex] cm at [latex]t[/latex] seconds after the jump.
- Sketch one period of the position function for [latex]t\ge 0[/latex].
- Find the velocity function.
- Sketch one period of the velocity function for [latex]t\ge 0[/latex].
- Determine the times when the velocity is 0 over one period.
- Find the acceleration function.
- Sketch one period of the acceleration function for [latex]t\ge 0[/latex].
29. The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 \sin x[/latex] where [latex]y[/latex] is the number of hamburgers sold and [latex]x[/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y^{\prime}[/latex] and determine the intervals where the number of burgers being sold is increasing.
30. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \cos t[/latex], where [latex]t[/latex] is the number of months since January. Find [latex]y^{\prime}[/latex] and use a calculator to determine the intervals where the amount of rain falling is decreasing.
For the following exercises (31-33), use the quotient rule to derive the given equations.
31. [latex]\frac{d}{dx}(\cot x)=−\csc^2 x[/latex]
32. [latex]\frac{d}{dx}(\sec x)= \sec x \tan x[/latex]
33. [latex]\frac{d}{dx}(\csc x)=−\csc x \cot x[/latex]
34. Use the definition of derivative and the identity
[latex]\cos (x+h)= \cos x \cos h- \sin x \sin h[/latex] to prove that [latex]\frac{d}{dx}(\cos x)=−\sin x[/latex].
For the following exercises (35-39), find the requested higher-order derivative for the given functions.
35. [latex]\frac{d^3 y}{dx^3}[/latex] of [latex]y=3 \cos x[/latex]
36. [latex]\frac{d^2 y}{dx^2}[/latex] of [latex]y=3 \sin x+x^2 \cos x[/latex]
37. [latex]\frac{d^4 y}{dx^4}[/latex] of [latex]y=5 \cos x[/latex]
38. [latex]\frac{d^2 y}{dx^2}[/latex] of [latex]y= \sec x+ \cot x[/latex]
39. [latex]\frac{d^3 y}{dx^3}[/latex] of [latex]y=x^{10}- \sec x[/latex]
