MA005: Calculus I

1. Let $\mathrm{f}(1)=2$ and $\mathrm{f}^{\prime}(1)=3$. Find the values of $\mathbf{D}\left(\mathrm{f}^{2}(\mathrm{x})\right), \operatorname{D}\left(\mathrm{f}^{5}(\mathrm{x})\right)$, and $\mathrm{D}(\sqrt{\mathrm{f}(\mathrm{x})})$ at $\mathrm{x}=1$.

3. Estimate the values of $f(x)$ and $f^{\prime}(x)$ in Fig. 5 and determine

(a) $\frac{d}{d x}\left(f^{2}(x)\right)$ at $x=1$ and $3$

(b) $\mathrm{D}\left(\mathrm{f}^{3}(\mathrm{x})\right)$

at $\mathrm{x}=1$ and $3$

(c) $\mathrm{D}\left(\mathrm{f}^{5}(\mathrm{x})\right)$ at $\mathrm{x}=1$ and $3$.

In problems 5-9, find the derivative of each function.

5. $f(x)=(2 x-8)^{5}$

7. $\mathrm{f}(\mathrm{x})=\mathrm{x} \cdot(3 \mathrm{x}+7)^{5}$

9. $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}^{2}+6 \mathrm{x}-1}$

11. A weight attached to a spring is at a height of $\mathrm{h}(\mathrm{t})=3-2 \sin (\mathrm{t})$ feet above the floor $\mathrm{t}$ seconds after it is released.

(a) Graph $\mathrm{h}(\mathrm{t})$

(b) At what height is the weight when it is released?

(c) How high does the weight ever get above the floor and how close to the floor does it ever get?

(d) Determine the height, velocity and acceleration at time t. (Be sure to include the correct units.)

(e) Why is this an unrealistic model of the motion of a weight on a real spring?

13. The kinetic energy $\mathrm{K}$ of an object of mass $\mathrm{m}$ and velocity $\mathrm{v}$ is $\frac{1}{2} \mathrm{~m} \cdot \mathrm{v}^{2}$.

(a) Find the kinetic energy of an object with mass $\mathrm{m}$ and height $\mathrm{h}(\mathrm{t})=5 \mathrm{t}$ feet at $\mathrm{t}=1$ and $2$ seconds.

(b) Find the kinetic energy of an object with mass $\mathrm{m}$ and height $\mathrm{h}(\mathrm{t})=\mathrm{t}^{2}$ feet at $t = 1$ and $2$ seconds.

In problems 15-19, find the derivatives $\mathbf{d f} / \mathbf{d} \mathbf{x}$.

15. $f(x)=x \cdot \sin (x)$

17. $f(x)=e^{x}-\sec (x)$

19. $f(x)=e^{-x}+\sin (x)$

In problems 21-25, find the equation of the line tangent to the graph of the function at the given point.

21. $f(x)=(x-5)^{7}$ at $(4,-1)$

23. $f(x)=\sqrt{25-x^{2}}$ at $(3,4)$

25. $\mathrm{f}(\mathrm{x})=(\mathrm{x}-\mathrm{a})^{5}$ at $(\mathrm{a}, 0)$

27. Find the equation of the line tangent to $f(x)=e^{x}$ at the point $\left(3, e^{3}\right)$. Where will this tangent line intersect the $\mathrm{x}$-axis? Where will the tangent line to $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{X}}$ at the point $\left(\mathrm{p}, \mathrm{e}^{\mathrm{p}}\right)$ intersect the $\mathrm{x}$-axis?

In problems 29-33, calculate $\mathbf{f}^{\prime}$ and $\mathbf{f}^{\prime \prime}$

29. $f(x)=\cos (x)$

31. $f(x)=x^{2} \cdot \sin (x)$

33. $f(x)=e^{x} \cdot \cos (x)$

35. What will the $2^{\text {nd }}$ derivative of a quadratic polynomial be? The $3^{\text {rd }}$ derivative? The $4^{\text {th }}$ derivative?

37. What can you say about the $\mathrm{n}^{\text {th }}$ and $(\mathrm{n}+1)^{\text {st }}$ derivatives of a polynomial of degree $\mathrm{n}$ ?

In problems 39-41, you are given $f^{\prime}$. Find a function $\mathrm{f}$ with the given derivative.

39. $\mathrm{f}^{\prime}(\mathrm{x})=5 \mathrm{e}^{\mathrm{x}}$

41. $f^{\prime}(x)=5\left(1+e^{x}\right)^{4} \cdot e^{x}$

43. The function $f(x)=\left\{\begin{array}{ll}x \cdot \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array} \quad\right.$ in Fig. 6 is continuous at 0 since $\lim\limits_{h \rightarrow 0} f(x)=0=f(0)$. Is $f$ differentiable at $0 ?$ (Use the definition of $f^{\prime}(0)$ and consider $\left.\lim\limits_{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} . \right.$ )

The number $e$ appears in a variety of unusual situations.

Problems 45 – 47 illustrate a few of them.

45. Use your calculator to examine the values of $\left(1+\frac{1}{\mathrm{x}}\right)^{\mathrm{x}}$ when $\mathrm{x}$ is relatively large, for example, $\mathrm{x}=100,1000$, and 10000 . Try some other large values for $\mathrm{x}$. If $\mathrm{x}$ is large, the value of $\left(1+\frac{1}{x}\right)^{x}$ is close to what number?

47. (a) Calculate the value of the sums $\mathrm{s}_{1}=1+\frac{1}{1 !}, \mathrm{s}_{2}=1+\frac{1}{1 !}+\frac{1}{2 !}, \mathrm{s}_{3}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}$, $\mathrm{s}_{4}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}, \mathrm{s}_{5}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\frac{1}{5 !}$, and $\mathrm{s}_{6}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\frac{1}{5 !}+\frac{1}{6 !}$

(b) What value do the sums in part (a) seem to be approaching? Calculate $s_{7}$ and $s_{8}$.

($\mathrm{n} !=$ product of all positive integers from 1 to n. For example, $2 !=1 \cdot 2=2,3 !=1 \cdot 2 \cdot 3=6,4 !=24 .$)

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.4-More-Differentiation-Problems.pdf
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