Practice Problems

1. Match the graphs of the three functions in Fig. 8 with the graphs of their derivatives.

In problems 3-5, find the slope $\mathrm{m}_{\mathrm{sec}}$ of the secant line through the two given points and then calculate $m_{\tan }=\lim _{h \rightarrow 0} \mathrm{~m}_{\mathrm{sec}}$.

3. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$

(a) $(-2,4),\left(-2+\mathrm{h},(-2+\mathrm{h})^{2}\right)$

(b) $(0.5,0.25),\left(0.5+\mathrm{h},(0.5+\mathrm{h})^{2}\right)$

5. $f(x)=7 x-x^{2}$

(a) $(1,6),\left(1+\mathrm{h}, 7(1+\mathrm{h})-(1+\mathrm{h})^{2}\right)$

(b) $\left(x, 7 x-x^{2}\right),\left(x+h, 7(x+h)-(x+h)^{2}\right)$

7. Use the graph in Fig. 10 to estimate the values of these limits. (It helps to recognize what the limit represents.)

(a) $\lim\limits_{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$

(b) $\lim\limits_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$

(c) $\lim\limits_{h \rightarrow 0} \frac{f(2+h)-1}{h}$

(d) $\lim\limits_{w \rightarrow 0} \frac{f(3+w)-f(3)}{w}$

(e) $\lim\limits_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}$

(f) $\lim\limits_{s \rightarrow 0} \frac{f(5+s)-f(5)}{s}$

In problems 9 – 11, use the Definition of the derivative to calculate $f '(x)$ and then evaluate $f '(3)$.

9. $f(x) = x^2 + 8$ 

11. $f(x) = 2x^3 – 5x$ 

13. Graph $f(x) = x , g(x) = x^2 + 3$ and $h(x) = x^2 – 5$. Calculate the derivatives of $f$, $g$, and $h$. 

In problems 15 – 17, find the slopes and equations of the lines tangent to $y = f(x)$ at the given points. 

15. $f(x) = x^2 + 8$ at $(1,9)$ and $(–2,12)$ 

17. $f(x) = sin(x)$ at $(π, 0)$ and $(π/2,1)$ 

19. (a) Find the equation of the line tangent to the graph of $y = x^2 + 1$ at the point $(2,5)$. 

(b) Find the equation of the line perpendicular to the graph of $y = x^2 + 1$ at $(2,5)$. 

(c) Where is the tangent to the graph of $y = x^2 + 1$ horizontal? 

(d) Find the equation of the line tangent to the graph of $y = x^2 + 1$ at the point $(p,q)$. 

(e) Find the point(s) $(p,q)$ on the graph of $y = x^2 + 1$ so the tangent line to the curve at $(p,q)$ goes through the point $(1, –7)$.

21. (a) Find the angle that the tangent line to $y = x^2$ at $(1,1)$ makes with the x–axis. 

(b) Find the angle that the tangent line to $y = x^3$ at $(1,1)$ makes with the x–axis. 

(c) The curves $y = x^2$ and $y = x^3$ intersect at the point $(1,1)$. Find the angle of intersection of the two curves (actually the angle between their tangent lines) at the point $(1,1)$. 

23. Fig. 13 shows the graph of the height of an object at time $t$. Sketch the graph of the object's upward velocity. What are the units for each axis on the velocity graph? 

25. A rock dropped into a deep hole will drop $d(x) = 16x^2$ feet in $x$ seconds. 

(a) How far into the hole will the rock be after 4 seconds? 5 seconds? 

(b) How fast will it be falling at exactly 4 seconds? 5 seconds? $x$ seconds? 

27. It costs $C(x) = √x$ dollars to produce $x$ golf balls. What is the marginal production cost to make a golf ball? What is the marginal production cost when $x = 25$? when $x= 100$? (Include units.) 

29. Define $A(x)$ to be the area bounded by the x–axis, the line $y = x$, and a vertical line at $x$ (Fig. 15). 

(a) Evaluate $A(0), A(1), A(2)$ and $A(3)$. 

(b) Find a formula which represents $A(x)$ for all $x ≥ 0: A(x) =$? 

(c) Determine $\frac{\mathrm{d} \mathrm{A}(\mathrm{x})}{\mathrm{dx}}$.

(d) What does $\frac{\mathrm{d} \mathrm{A}(\mathrm{x})}{\mathrm{dx}}$ represent?

31. Find (a) $D\left(x^{9}\right)$

(b) $\frac{\mathbf{d} x^{2 / 3}}{d x}$

(c) $D\left(\frac{1}{x^{4}}\right)$

(d) $\mathrm{D}\left(\mathrm{x}^{\pi}\right)$

(e) $\frac{\mathbf{d}|\mathrm{x}+5|}{\mathbf{d x}}$

In problems 31 – 37, find a function $f$ which has the given derivative. (Each problem has several correct answers, just find one of them.) 

33. $f^{\prime}(x)=3 x^{2}+8 x$

35. $\frac{\mathbf{d} \mathrm{f}(\mathrm{t})}{\mathrm{dt}}=5 \cos (\mathrm{t})$

37. $\mathrm{D}(\mathrm{f}(\mathrm{x}))=\mathrm{x}+\mathrm{x}^{2}$

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.2-Definition-of-Derivative.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. (a) derivative of $g$

(b) derivative of $\mathrm{h}$

(c) derivative of $\mathrm{f}$

3. (a) $\mathrm{m}_{\mathrm{sec}}=\mathrm{h}-4, \mathrm{~m}_{\tan }=\lim\limits_{h \rightarrow 0} \mathrm{~m}_{\mathrm{sec}}=-4$.

(b) $\mathrm{m}_{\mathrm{sec}}=\mathrm{h}+1, \mathrm{~m}_{\tan }=\lim\limits_{h \rightarrow 0} \mathrm{~m}_{\mathrm{sec}}=1$.

5. (a) $\mathrm{m}_{\mathrm{sec}}=5-\mathrm{h}, \mathrm{m}_{\tan }=\lim\limits_{h \rightarrow 0} \mathrm{~m}_{\mathrm{sec}}=5$

(b) $\mathrm{m}_{\mathrm{sec}}=7-2 \mathrm{x}-\mathrm{h}, \mathrm{m}_{\tan }=\lim\limits_{h \rightarrow 0} \mathrm{~m}_{\mathrm{sec}}=7-2 \mathrm{x}$.

7. $-1$

(b) $-1$

(c) $0$

(d) $+1$

(e) DNE

(f) DNE

9. $\mathrm{f}^{\prime}(\mathrm{x})=\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \rightarrow 0} \frac{\left\{(x+h)^{2}+8\right\}-\left\{x^{2}+8\right\}}{h}=\lim\limits_{h \rightarrow 0} \frac{2 x h+h^{2}}{h}=\lim\limits_{h \rightarrow 0} 2 x+h=2 x \cdot \mathrm{f}^{\prime}(3)=6$.

11. $\mathrm{f}^{\prime}(\mathrm{x})=\lim\limits_{h \rightarrow 0} \frac{\left\{2(x+h)^{3}-5(x+h)\right\}-\left\{2 x^{3}-5 x\right\}}{h}=\lim\limits_{h \rightarrow 0} \frac{6 x^{2} h+6 x h^{2}+2 h^{3}-5 h}{h}=6 x^{2}-5 . \mathrm{f}^{\prime}(3)=49$.

13. For any constant $\mathrm{C}$, if $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{C}$, then

$\mathrm{f}^{\prime}(\mathrm{x})==\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \rightarrow 0} \frac{\left\{(x+h)^{2}+C\right\}-\left\{x^{2}+C\right\}}{h}=\lim\limits_{h \rightarrow 0} \frac{2 x h+h^{2}}{h}=\lim\limits_{h \rightarrow 0} 2 \mathrm{x}+\mathrm{h}=2 \mathrm{x}$

The graphs of $f(x) = x^2, g(x) = x^2 +3$ and $h(x) = x^2 – 5$ are "parallel" parabolas: $g$ is $f$ shifted up 3 units, and $h$ is $f$ shifted down 5 units.

15. $f '(x) = 2x$. Then $f '(1) = 2$ and the equation of the tangent line at $(1,9)$ is $y – 9 = 2(x – 1)$ or $y = 2x + 7$.        

$f '(–2) = – 4$ and the equation of the tangent line at $(–2,12)$ is $y – 12 = –4(x + 2)$ or $y = –4x + 4$.

17. $f '(x) = cos(x)$. Then $f '(π) = cos(π) = –1$ and the equation of the tangent line at $(π,0)$ is $y – 0 = –1(x – π)$ or $y = –x + π$. 

$f '(π/2) = cos(π/2) = 0$ and the equation of the tangent line at $(π/2,1)$ is $y – 1 = 0(x – π/2)$ or $y = 1$.

19. (a) $y – 5 = 4(x – 2)$ or $y = 4x – 3 (b) x + 4y = 22$ or $y = –0.25x + 5.5$ 

(c) $f '(x) = 2x$ so the tangent line is horizontal when $x = 0$: at the point $(0,1)$.

(d) $f '(p) = 2p$ (the slope of the tangent line) so $y – q = 2p(x – p)$ or $y = 2px + (q – 2p^2)$. 

Since $q = p^2 + 1$, the equation of the tangent line becomes $y = 2px + (p^2 + 1 – 2p^2) = 2px – p^2 + 1$. 

(e) We need $p$ such that $–7 = 2p(1) – p2 + 1 or p2 – 2p – 8 = 0$. Then $p = –2, 4$. There are two points with the property we want: $(–2, 5)$ and $(4, 17)$.

21. (a) $y ' = 2x$, so when $x = 1, y ' = 2$. Angle $= arctan(2) ≈ 1.107$ radians $≈ 63°$. 

(b) $y ' = 3x^2$, so when $x = 1, y ' = 3$. Angle $= arctan(3) ≈ 1.249$ radians $≈ 72°$. 

(c) Angle $≈ 1.249 – 1.107$ radians $= 0.142$ radians (or  angle $= 72° – 63° = 9°$)

23. Graph. On the graph of upward velocity, the units on the horizontal axis are "seconds" and the units on the vertical axis are "feet per second".

25. (a) $d(4) = 256$ ft. $d(5) = 400$ ft.  (b) $d '(x) = 32x$  $d '(4) = 128$ ft/sec $d '(5) = 160$ ft/sec.

27. $C(x) = √x$ dollars to produce $x$ golf balls.

Marginal production cost is $\mathrm{C}^{\prime}(\mathrm{x})=\frac{1}{2 \sqrt{\mathrm{x}}} \quad$ dollars per golf ball. $\mathrm{C}^{\prime}(25)=\frac{1}{2 \sqrt{25}}=\frac{1}{10}$ dollars per golf ball. $\mathrm{C}^{\prime}(100)=\frac{1}{2 \sqrt{100}}=\frac{1}{20}$ dollars per golf ball.

29.

(a) $\mathrm{A}(0)=0, \mathrm{~A}(1)=1 / 2, \mathrm{~A}(2)=2$ and $\mathrm{A}(3)=9 / 2$

(b) $\mathrm{A}(\mathrm{x})=\mathrm{x}^{2} / 2(\mathrm{x} \geq 0)$

(c) $\frac{\mathrm{d} \mathrm{A}(\mathrm{x})}{\mathrm{dx}}=\mathrm{x}$.

(d) $\frac{\mathrm{d} \mathrm{A}(\mathrm{x})}{\mathrm{dx}}$ represents the rate at which $\mathrm{A}(\mathrm{x})$ is increasing, the rate at which area is accumulating.

31 .

(b) $\frac{2}{3 \mathrm{x}^{1 / 3}}$

(c) $\frac{-4}{x^{5}}$

(d) $\pi \mathrm{x}^{\pi-1}$

(e) 1 if $x>-5$ and $-1$ if $x$

33. $f(x)=x^{3}+4 x^{2}$ (plus any constant)

35.$f(x)=5 \cdot \sin (t)$

37. $f(x)=\frac{1}{2} x^{2}+\frac{1}{3} x^{3}$