3.2E: Exercises for Section 3.2

In exercises 1 - 7, find \(f'(x)\) for each function.

1) \(f(x)=3x\left(18x^4+\dfrac{13}{x+1}\right)\)

Answer

\(f'(x) = 270x^4+\dfrac{39}{(x+1)^2}\)

2) \(f(x)=(x+2)(2x^2−3)\)

3) \(f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)\)

Answer

\(f'(x) = \dfrac{−5}{x^2}\)

4) \(f(x)=\dfrac{x^3+2x^2−4}{3}\)

5) \(f(x)=\dfrac{4x^3−2x+1}{x^2}\)

Answer

\(f'(x) = \dfrac{4x^4+2x^2−2x}{x^4}\)

6) \(f(x)=\dfrac{x^2+4}{x^2−4}\)

7) \(f(x)=\dfrac{x+9}{x^2−7x+1}\)

Answer

\(f'(x) = \dfrac{−x^2−18x+64}{(x^2−7x+1)^2}\)

In exercises 8 - 11, assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of each of the functions \(h(x)\).

8) \(h(x)=4f(x)+\dfrac{g(x)}{7}\)

9) \(h(x)=x^3f(x)\)

Answer

\(h'(x)=3x^2f(x)+x^3f′(x)\)

10) \(h(x)=\dfrac{f(x)g(x)}{2}\)

11) \(h(x)=\dfrac{3f(x)}{g(x)+2}\)

Answer

\(h'(x)=\dfrac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}\)

For exercises 12 - 15, assume that \(f(x)\) and \(g(x)\) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

\(x\)	1	2	3	4
\(f(x)\)	3	5	−2	0
\(g(x)\)	2	3	−4	6
\(f′(x)\)	−1	7	8	−3
\(g′(x)\)	4	1	2	9

12) Find \(h′(1)\) if \(h(x)=x f(x)+4g(x)\).

13) Find \(h′(2)\) if \(h(x)=\dfrac{f(x)}{g(x)}\).

Answer

\(h'(2) =\frac{16}{9}\)

14) Find \(h′(3)\) if \(h(x)=2x+f(x)g(x)\).

15) Find \(h′(4)\) if \(h(x)=\dfrac{1}{x}+\dfrac{g(x)}{f(x)}\).

Answer

\(h'(4)\) is undefined.

In exercises 16 - 18, use the following figure to find the indicated derivatives, if they exist.

16) Let \(h(x)=f(x)+g(x)\). Find

a) \(h′(1)\),

b) \(h′(3)\), and

c) \(h′(4)\).

17) Let \(h(x)=f(x)g(x).\) Find

a) \(h′(1),\)

b) \(h′(3)\), and

c) \(h′(4).\)

Answer

a. \(h'(1) = 2\),
b. \(h'(3)\) does not exist,
c. \(h'(4) = 2.5\)

18) Let \(h(x)=\dfrac{f(x)}{g(x)}.\) Find

a) \(h′(1),\)

b) \(h′(3)\), and

c) \(h′(4).\)

19) Find the equation of the tangent line to the graph of \(f(x)=x^2+\dfrac{4}{x}−10\) at \(x=8\).

20) Find the equation of the tangent line to the graph of \(f(x)=(3x−x^2)(3−x−x^2)\) at \(x=1\).

Answer

\(y=−5x+7\)

21) Find the equation of the line containing the point \(P(3,3)\) and tangent to the graph of \(f(x)=\dfrac{6}{x−1}\).

Answer

\(y=−\frac{3}{2}x+\frac{15}{2}\)

22) Determine all points on the graph of \(f(x)=x^3+x^2−x−1\) for which the slope of the tangent line is

a. horizontal

b. −1.

23) Find a quadratic polynomial such that \(f(1)=5,\; f′(1)=3\) and \(f''(1)=−6.\)

Answer

\(y=−3x^2+9x−1\)

24) A car driving along a freeway with traffic has traveled \(s(t)=t^3−6t^2+9t\) meters in \(t\) seconds.

a. Determine the time in seconds when the velocity of the car is 0.

b. Determine the acceleration of the car when the velocity is 0.

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