MA005: Calculus I

1. The graph of $y = f(x)$ is given in Fig. 6. 

(a) At which integers is $f$ continuous?  

(b) At which integers is $f$ differentiable? 

3. Use the values given in the table to determine the values of $f.g, D( f.g ), f/g$ and $D( f/g )$. 

7. Calculate $D( (x – 5)(3x + 7) )$ by (a) using the product rule and (b) expanding the product and then differentiating. Verify that both methods give the same result. 

9. Calculate $\frac{\mathbf{d}}{\mathbf{d x}}\left(\frac{\cos (\mathrm{x})}{\mathrm{x}^{2}}\right)$.

11. Calculate $\mathbf{D}\left(\sin ^{2}(\mathrm{x})\right)$ and $\mathbf{D}\left(\cos ^{2}(\mathrm{x})\right)$.

13. Find values for the constants $a, b$ and $c$ so that the parabola $f(x)=a x^{2}+b x+c$ has $f(0)=0$, $f^{\prime}(0)=0$ and $f^{\prime}(10)=30$

15. If $\mathrm{f}$ and $\mathrm{g}$ are differentiable functions which always differ by a constant $(\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})=\mathrm{k}$ for all $x$ ), then what can you conclude about their graphs and their derivatives?

17. If the product of $f$ and $g$ is a constant $(\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x})=\mathrm{k}$ for all $\mathrm{x})$, then how are $\frac{\mathrm{D}(\mathrm{f}(\mathrm{x}))}{\mathrm{f}(\mathrm{x})}$ and $\frac{\mathrm{D}(\mathrm{g}(\mathrm{x}))}{\mathrm{g}(\mathrm{x})}$ related?

19. $f(x)=x^{2}-5 x+13$

21. $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-2 \cos (\mathrm{x})$

23. $f(x)=x^{3}+9 x^{2}+6$

25. $f(x)=x^{3}+2 x^{2}+2 x-1$

27. $\mathrm{f}(\mathrm{x})=\mathrm{x} \cdot \sin (\mathrm{x})$ and $0 \leq \mathrm{x} \leq 5$. (You may need to use the Bisection Algorithm or the "trace" option on a calculator to approximate where $\mathrm{f}^{\prime}(\mathrm{x})=0$.)

29. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{A} \mathrm{x}^{2}+\mathrm{B} \mathrm{x}+\mathrm{C}$ with constants $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$. Can you find conditions on the constants $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ which will guarantee that the graph of $\mathrm{y}=\mathrm{f}(\mathrm{x})$ has two distinct "vertices"? (Here a "vertex" means a place where the curve changes from increasing to decreasing or from decreasing to increasing.)

Where are the functions in problems 30-37 differentiable?

31. $f(x)=\frac{x-5}{x+3}$

33. $f(x)=\frac{x^{2}+x}{x^{2}-3 x}$

35. $\mathrm{f}(\mathrm{x})=\left|\mathrm{x}^{3}-1\right|$

37. $f(x)= \begin{cases}x & \text { if } x < 0 \\ \sin (x) & \text { if } x \geq 0\end{cases}$

39. For what values of $\mathrm{A}$ and $\mathrm{B}$ is $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\mathrm{Ax}+\mathrm{B} & \text { if } \mathrm{x} < 1 \\ \mathrm{x}^{2}+\mathrm{x} & \text { if } \mathrm{x} \geq 1\end{array}\right.$ differentiable at $\mathrm{x} = 1$ ?

41. If an arrow is shot straight up from ground level on the moon with an initial velocity of 128 feet per second, its height will be $\mathrm{h}(\mathrm{x})=-2.65 \mathrm{x}^{2}+128 \mathrm{x}$ feet at $\mathrm{x}$ seconds. Do parts (a) - (e) of problem 40 using this new equation for $h$.

43. If an object on Earth is propelled upward from ground level with an initial velocity of $\mathrm{v}_{0}$ feet per second, then its height at $\mathrm{x}$ seconds will be $h(x)=-16 x^{2}+v_{0} x$

(a) What will be the object's velocity after $\mathrm{x}$ seconds?

(b) What is the greatest height the object will reach? 

(c) How long will the object remain aloft? 

45. The best high jumpers in the world manage to lift their centers of mass approximately 6.5 feet.  

(a) What is the initial vertical velocity these high jumpers attain? 

(b) How long are these high jumpers in the air? 

(c) With the initial velocity in part (a), how high would they lift their centers of mass on the moon?

47. (a) Find the equation of the line $L$ which is tangent to the curve $y=\frac{1}{x}$ at the point $\left(2, \frac{1}{2}\right)$.

(b) Graph $\mathrm{y}=1 / \mathrm{x}$ and $\mathrm{L}$ and determine where $\mathrm{L}$ intersects the $\mathrm{x}$-axis and the $\mathrm{y}$-axis.

(c) Determine the area of the region in the first quadrant bounded by $\mathrm{L}$, the $\mathrm{x}$-axis and the $\mathrm{y}$-axis.

49. Find values for the coefficients $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ so that the parabola $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ goes through the point $(1,4)$ and is tangent to the line $\mathrm{y}=9 \mathrm{x}-13$ at the point $(3,14)$.

51. (a) Find a function $\mathrm{f}$ so that $\mathrm{D}(\mathrm{f}(\mathrm{x}))=3 \mathrm{x}^{2}$.

(b) Find another function $\mathrm{g}$ so that $\mathrm{D}(\mathrm{g}(\mathrm{x}))=3 \mathrm{x}^{2}$.

(c) Can you find more functions whose derivatives are $3 \mathrm{x}^{2}$ ?

53. The graph of $y=f^{\prime}(x)$ is given in Fig. 12.

(a) Assume $\mathrm{f}(0)=0$ and sketch the graph of $\mathrm{y}=\mathrm{f}(\mathrm{x})$.

(b) Assume $\mathrm{f}(0)=1$ and graph $\mathrm{y}=\mathrm{f}(\mathrm{x})$.

55. Assume that $\mathrm{f}$ and $\mathrm{g}$ are differentiable functions and that $\mathrm{g}(\mathrm{x}) \neq 0$. State why each step in the following proof of the Quotient Rule is valid.

$\begin{align*} \begin{aligned} &\operatorname{D}\left(\frac{\mathrm{f}(x)}{g(x)}\right)=\lim _{h \rightarrow 0} \frac{1}{h}\left\{\frac{f(x+h)}{g(x+h)}-\frac{f(x)}{g(x)}\right\}=\lim _{h \rightarrow 0} \frac{1}{h}\left\{\frac{f(x+h) g(x)-g(x+h) f(x)}{g(x+h) g(x)}\right\} \\ &=\lim _{h \rightarrow 0} \frac{1}{g(x+h) g(x)}\left\{\frac{f(x+h) g(x)+(-f(x) g(x)+f(x) g(x))-g(x+h) f(x)}{h}\right\} \\ &=\lim _{h \rightarrow 0} \frac{1}{g(x+h) g(x)}\left\{g(x) \frac{f(x+h)-f(x)}{h}-f(x) \frac{g(x+h)-g(x)}{h}\right\} \\ &=\frac{1}{g^{2}(x)}\left\{g(x) \cdot f^{\prime}(x)-f(x) \cdot g^{\prime}(x)\right\}=\frac{g(x) \cdot f^{\prime}(x)-f(x) \cdot g^{\prime}(x)}{g^{2}(x)} \end{aligned} \end{align*}$

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.3-Derivatives-Properties-and-Formulas.pdf
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