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# 2.4E: Exercises

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### Exercise $$\PageIndex{1}$$

For the following exercises, find $$\frac{dy}{dx}$$ for the given functions:

1) $$y=x^2−secx+1$$

2) $$y=3cscx+\frac{5}{x}$$

3) $$y=x^2cotx$$

4) $$y=x−x^3sinx$$

5) $$y=\frac{secx}{x}$$

6) $$y=sinxtanx$$

7) $$y=(x+cosx)(1−sinx)$$

8) $$y=\frac{tanx}{1−secx}$$

9) $$y=\frac{1−cotx}{1+cotx}$$

10) $$y=cosx(1+cscx)$$

1. $$\frac{dy}{dx}=2x−secxtanx$$

3. $$\frac{dy}{dx}=2xcotx−x^2csc^2x$$

5. $$\frac{dy}{dx}=\frac{xsecxtanx−secx}{x^2}$$

7. $$\frac{dy}{dx}=(1−sinx)(1−sinx)−cosx(x+cosx)$$

9. $$\frac{dy}{dx}=\frac{2csc^2x}{(1+cotx)^2}$$

### Exercise $$\PageIndex{2}$$

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of $$x$$.

1) $$f(x)=−sinx,x=0$$

2) $$f(x)=cscx,\,x=\frac{π}{2}$$

3) $$f(x)=1+cosx,\,x=\frac{3π}{2}$$

4) $$f(x)=secx,\,x=\frac{π}{4}$$

5) $$f(x)=x^2−tanx, \,x=0$$

6) $$f(x)=5cot(x), \,x=\frac{π}{4}$$

1. $$y=−x$$ 3. $$y=x+\frac{2−3π}{2}$$ 5. $$y=x+\frac{2−3π}{2}$$ ### Exercise $$\PageIndex{3}$$

For the following exercises, find $$\frac{d^2y}{dx^2}$$ for the given functions.

1) $$y=xsinx−cosx$$

2) $$y=sinxcosx$$

3) $$y=x−\frac{1}{2}sinx$$

4) $$y=\frac{1}{x}+tanx$$

5) $$y=2cscx$$

6) $$y=sec^2x$$

1. $$3cosx−xsinx$$

3. $$\frac{1}{2}sinx$$

5. $$csc(x)(3csc^2(x)−1+cot^2(x))$$

### Exercise $$\PageIndex{4}$$

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

$$\frac{(2n+1)π}{4}$$,where $$n$$ is an integer.

### Exercise $$\PageIndex{5}$$

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

Under Construction

### Exercise $$\PageIndex{6}$$

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

$$(\frac{π}{4},1),(\frac{3π}{4},−1)$$

### Exercise $$\PageIndex{7}$$

A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $$s(t)=−6cost$$ 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.

Under Construction

### Exercise $$\PageIndex{8}$$

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

$$a=0,b=3$$

### Exercise $$\PageIndex{9}$$

After a diver jumps off a diving board, the edge of the board oscillates with position given by $$s(t)=−5cost$$ 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 $$t≥0$$.

Under Construction

### Exercise $$\PageIndex{10}$$

The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $$y=10+5sinx$$ 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.

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

### Exercise $$\PageIndex{11}$$

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

Under Construction

### Exercise $$\PageIndex{12}$$

For the following exercises, use the quotient rule to derive the given equations.

1) $$\frac{d}{dx}(cotx)=−csc^2x$$

2) $$\frac{d}{dx}(secx)=secxtanx$$

3) $$\frac{d}{dx}(cscx)=−cscxcotx$$

Under Construction

### Exercise $$\PageIndex{13}$$

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

1. $$\frac{d^3y}{dx^3}$$ of $$y=3cosx$$
2. $$\frac{d^2y}{dx^2}$$ of $$y=3sinx+x^2cosx$$
3. $$\frac{d^4y}{dx^4}$$ of $$y=5cosx$$
4. $$\frac{d^2y}{dx^2}$$ of $$y=secx+cotx$$
5. $$\frac{d^3y}{dx^3}$$ of $$y=x^{10}−secx$$
1. $$3sinx$$
3. $$5cosx$$
5. $$720x^7−5tan(x)sec^3(x)−tan^3(x)sec(x)$$