$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.5E: Exercises for Section 3.5

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

In exercises 1 - 10, find $$\dfrac{dy}{dx}$$ for the given functions.

1) $$y=x^2−\sec x+1$$

$$\dfrac{dy}{dx}=2x−\sec x\tan x$$

2) $$y=3\csc x+\dfrac{5}{x}$$

3) $$y=x^2\cot x$$

$$\dfrac{dy}{dx}=2x\cot x−x^2\csc^2 x$$

4) $$y=x−x^3\sin x$$

5) $$y=\dfrac{\sec x}{x}$$

$$\dfrac{dy}{dx}=\dfrac{x\sec x\tan x−\sec x}{x^2}$$

6) $$y=\sin x\tan x$$

7) $$y=(x+\cos x)(1−\sin x)$$

$$\dfrac{dy}{dx}=(1−\sin x)(1−\sin x)−\cos x(x+\cos x)$$

8) $$y=\dfrac{\tan x}{1−\sec x}$$

9) $$y=\dfrac{1−\cot x}{1+\cot x}$$

$$\dfrac{dy}{dx}=\dfrac{2\csc^2 x}{(1+\cot x)^2}$$

10) $$y=(\cos x)(1+\csc x)$$

In exercises 11 - 16, find the equation of the tangent line to each of the given functions at the indicated values of $$x$$. 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] $$f(x)=−\sin x,\quad x=0$$

$$y=−x$$ 12) [T] $$f(x)=\csc x,\quad x=\frac{π}{2}$$

13) [T] $$f(x)=1+\cos x,\quad x=\frac{3π}{2}$$

$$y=x+\frac{2−3π}{2}$$ 14) [T] $$f(x)=\sec x,\quad x=\frac{π}{4}$$

15) [T] $$f(x)=x^2−\tan x, \quad x=0$$

$$y=−x$$ 16) [T] $$f(x)=5\cot x, \quad x=\frac{π}{4}$$

In exercises 17 - 22, find $$\dfrac{d^2y}{dx^2}$$ for the given functions.

17) $$y=x\sin x−\cos x$$

$$\dfrac{d^2y}{dx^2} = 3\cos x−x\sin x$$

18) $$y=\sin x\cos x$$

19) $$y=x−\frac{1}{2}\sin x$$

$$\dfrac{d^2y}{dx^2} = \frac{1}{2}\sin x$$

20) $$y=\dfrac{1}{x}+\tan x$$

21) $$y=2\csc x$$

$$\dfrac{d^2y}{dx^2} = \csc(x)(3\csc^2(x)−1+\cot^2(x))$$

22) $$y=\sec^2 x$$

23) Find all $$x$$ values on the graph of $$f(x)=−3\sin x\cos x$$ where the tangent line is horizontal.

$$x = \dfrac{(2n+1)π}{4}$$,where $$n$$ is an integer

24) Find all $$x$$ values on the graph of $$f(x)=x−2\cos x$$ for $$0<x<2π$$ where the tangent line has slope 2.

25) Let $$f(x)=\cot x.$$ Determine the points on the graph of $$f$$ for $$0<x<2π$$ where the tangent line(s) is (are) parallel to the line $$y=−2x$$.

$$\left(\frac{π}{4},1\right),\quad \left(\frac{3π}{4},−1\right),\quad\left(\frac{5π}{4},1\right),\quad \left(\frac{7π}{4},−1\right)$$

26) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $$s(t)=−6\cos t$$ where s is measured in inches and $$t$$ is measured in seconds. Find the rate at which the spring is oscillating at $$t=5$$ s.

27) Let the position of a swinging pendulum in simple harmonic motion be given by $$s(t)=a\cos t+b\sin t$$. Find the constants $$a$$ and $$b$$ such that when the velocity is 3 cm/s, $$s=0$$ and $$t=0$$.

$$a=0,\quad b=3$$

28) After a diver jumps off a diving board, the edge of the board oscillates with position given by $$s(t)=−5\cos t$$ cm at $$t$$ seconds after the jump.

a. Sketch one period of the position function for $$t≥0$$.

b. Find the velocity function.

c. Sketch one period of the velocity function for $$t≥0$$.

d. Determine the times when the velocity is $$0$$ over one period.

e. Find the acceleration function.

f. Sketch one period of the acceleration function for $$t≥0$$.

29) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $$y=10+5\sin x$$ where $$y$$ is the number of hamburgers sold and $$x$$ represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find $$y'$$ and determine the intervals where the number of burgers being sold is increasing.

$$y′=5\cos(x)$$, increasing on $$\left(0,\frac{π}{2}\right),\;\left(\frac{3π}{2},\frac{5π}{2}\right)$$, and $$\left(\frac{7π}{2},12\right)$$

30) [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by $$y(t)=0.5+0.3\cos t$$, where $$t$$ is months since January. Find $$y′$$and use a calculator to determine the intervals where the amount of rain falling is decreasing.

For exercises 31 - 33, use the quotient rule to derive the given equations.

31) $$\dfrac{d}{dx}(\cot x)=−\csc^2x$$

32) $$\dfrac{d}{dx}(\sec x)=\sec x\tan x$$

33) $$\dfrac{d}{dx}(\csc x)=−\csc x\cot x$$

34) Use the definition of derivative and the identity $$\cos(x+h)=\cos x\cos h−\sin x\sin h$$ to prove that $$\dfrac{d}{dx}(\cos x)=−\sin x$$.

For exercises 35 - 39, find the requested higher-order derivative for the given functions.

35) $$\dfrac{d^3y}{dx^3}$$ of $$y=3\cos x$$

$$\dfrac{d^3y}{dx^3} = 3\sin x$$

36) $$\dfrac{d^2y}{dx^2}$$ of $$y=3\sin x+x^2\cos x$$

37) $$\dfrac{d^4y}{dx^4}$$ of $$y=5\cos x$$

$$\dfrac{d^4y}{dx^4} = 5\cos x$$

38) $$\dfrac{d^2y}{dx^2}$$ of $$y=\sec x+\cot x$$

39) $$\dfrac{d^3y}{dx^3}$$ of $$y=x^{10}−\sec x$$

$$\dfrac{d^3y}{dx^3} = 720x^7−5\tan(x)\sec^3(x)−\tan^3(x)\sec(x)$$