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# 2E: Exercises

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## Chapter Review Exercises

#### Exercise $${1}$$

Let $$f(x+y)=f(x)+f(y)+x^2y+xy^2$$ and $$\lim_{x \rightarrow 0} \dfrac{f(x)}{x}=1$$. Then find $$f^'(x)$$.

$$1+x^2$$

#### Exercise $${2}$$

Use the limit definition of the derivative to find $$f'(x)$$, where $$f(x)=3x^2$$.
(You may use differentiation rules to check your answer). Find the tangent line to $$f(x)=3x^2$$ at $$x=1.$$

Under Construction

#### Exercise $${3}$$

Using the definition of the derivative (Newton's quotient)
find $$f'(x)$$ for
$$\displaystyle{f(x)=\frac{1}{\sqrt{2x+5}}}$$

Under Construction

#### Exercise $${4}$$

Use the limit denition of the derivative to find $$f'(x),$$ where $$f(x) = \frac{1}{x+1}$$

(You may using differentiation rules to check your answer). Then find the equation of

the tangent line to $$f(x)$$ at $$x = 2$$

Under Construction

#### Exercise $${5}$$

Find the derivatives of the following functions:

1) $$\displaystyle{f(x)=\sqrt{(x^2+x+1)}+{x^2}}$$

2) $$\displaystyle{g(x)=\frac{\cos x}{x+ \sin x}}$$

3) $$\displaystyle{h(t)=t^2+ \csc t+ +\sec^2 t}$$

4) $$\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}$$

5) $$f(x) =\displaystyle \frac{(x^2+1) \cot x}{3-\cos x \csc x}$$

6) $$\displaystyle{h(t)=\sqrt{\tan(2t-1)}}$$

7) $$\displaystyle{y(x)=\frac{x}{x^2-1}}$$

8) $$\displaystyle{f(x)=\frac{ln(x^2+1)}{x^2}}$$

9) $$f(x) = \left( \displaystyle \frac{x-5}{2x+1} \right)^3$$

10) $$\displaystyle{h(t)=\sin^2(\sqrt{\tan(2t-1)})}$$

11) $$\displaystyle{h(t)=\frac{\sin(t)}{t}}$$

12) $$y= \csc\left(\cot(\sqrt{4x})\right)$$

13) $$\displaystyle{y(x)=(x+5)^7(x^2-1)}$$

14) $$g(x)=\displaystyle \frac{\tan (2x) + \sqrt{x}}{\cos (x)+x^2}$$

15) $$g(x)=\displaystyle \frac{\cos (x) - \sqrt{x}}{\tan (x)+3x^2}$$

16) $$y=\displaystyle{\frac{3x+4}{x^2+5}}$$

17) $$y=\displaystyle{\sec(x)\tan(x)}$$

18) $$\displaystyle{y = \sqrt{5+\sqrt{6x}}}$$.

19) $$y= x^3 cos(x).$$

20) $$f(x)-\frac{tan(2x)}{2x +1}$$

Under Construction

#### Exercise $${6}$$

Find equations of the tangent and normal to $$y= \frac{2}{3-4 \sqrt{x}}$$ at the point $$(1,-2)$$.

Under Construction

#### Exercise $${7}$$

An object is released from rest (its initial velocity is zero) from the Empire State Building at a height of $$1250$$ ft above street level. The height of the object can

be modeled by the position functions $$s = f(t) = 1250 - 16t^2$$.

a. Verify the object is still falling at $$t = 5s$$:

b. Find the object's instantaneous velocity at the time $$t = 5s$$

Under Construction

#### Exercise $${8}$$

True or False? Justify the answer with a proof or a counterexample.

1) Every function has a derivative.

2) A continuous function has a continuous derivative.

3) A continuous function has a derivative.

4) If a function is differentiable, it is continuous.

1. False.

3. False.

#### Exercise $${9}$$

Use the limit definition of the derivative to exactly evaluate the derivative:

1) $$f(x)=\sqrt{x+4}$$

2) $$f(x)=\frac{3}{x}$$

1. $$\frac{1}{2\sqrt{x+4}}$$

#### Exercise $${10}$$

Find the derivatives of the following functions:

1) $$f(x)=3x^3−\frac{4}{x^2}$$

2) $$f(x)=(4−x^2)^3$$

3) $$f(x)=e^{sinx}$$

4) $$f(x)=ln(x+2)$$

5) $$f(x)=x^2cosx+xtan(x)$$

6) $$f(x)=\sqrt{3x^2+2}$$

7) $$f(x)=\frac{x}{4}sin^{−1}(x)$$

8) $$x^2y=(y+2)+xysin(x)$$

1. $$9x^2+\frac{8}{x^3}$$

3. $$e^{sinx}cosx$$

5. $$xsec^2(x)+2xcos(x)+tan(x)−x^2sin(x)$$

7. $$\frac{1}{4}(\frac{x}{\sqrt{1−x^2}}+sin^{−1}(x))$$

#### Exercise $${11}$$

Find the following derivatives of various orders:

1) First derivative of $$y=xln(x)cosx$$

2) Third derivative of $$y=(3x+2)^2$$

3) Second derivative of $$y=4^x+x^2sin(x)$$

1. $$cosx⋅(lnx+1)−xln(x)sinx$$

3. $$4^x(ln4)^2+2sinx+4xcosx−x^2sinx$$

#### Exercise $${12}$$

Find the equation of the tangent line to the following equations at the specified point:

1) $$y=cos^{−1}(x)+x$$ at $$x=0$$

2) $$y=x+e^x−\frac{1}{x}$$ at $$x=1$$

2. $$T=(2+e)x−2$$

#### Exercise $${13}$$

Draw the derivative for the following graphs.

21)   #### Exercise $${14}$$

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by $$w(t)=1.9+2.9cos(\frac{π}{6}t),$$ where t is measured in hours after midnight, and the height is measured in feet.

1) Find and graph the derivative. What is the physical meaning?

2) Find $$w′(3).$$ What is the physical meaning of this value?

2. $$w′(3)=−\frac{2.9π}{6}$$. At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

#### Exercise $${15}$$

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

 Hours after Midnight, August 26 Wind Speed (mph) 1 45 5 75 11 100 29 115 49 145 58 175 73 155 81 125 85 95 107 35

Wind Speeds of Hurricane KatrinaSource: http://news.nationalgeographic.com/n..._timeline.html.

1) Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?

2) Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

2. $$−7.5.$$ The wind speed is decreasing at a rate of 7.5 mph/hr