## Educators

FL   ### Problem 1

(a) How is the number $e$ defined?
(b) Use a calculator to estimate the values of the limits

$\displaystyle \lim_{h\to 0}\frac {2.7^h - 1}{h}$ and $\displaystyle \lim_{h\to 0}\frac {2.8^h - 1}{h}$

correct to two decimal places. What can you conclude about the value of $e$? Clarissa N.

### Problem 2

(a) Sketch, by hand, the graph of the function $f(x) = e^x,$ paying particular attention to the graph crosses the y-axes.
What fact allows you to do this?
(b) What types of functions are $f(x) = e^x$ and $g(x) = x^e?$
Compare the differentiation formulas for $f$ and $g.$
(c) Which of the two functions in part (b) grows more rapidly when $x$ is large? Clarissa N.

### Problem 3

Differentiate the function.
$f(x) = 2^{40}$ Clarissa N.

### Problem 4

Differentiate the function.
$f(x) = e^5$ Clarissa N.

### Problem 5

Differentiate the function.
$f(x) = 5.2x + 2.3$ Clarissa N.

### Problem 6

Differentiate the function.
$g(x) = \frac{7}{4}x^2 - 3x + 12$ Clarissa N.

### Problem 7

Differentiate the function.
$f(t) = 2t^3 - 3t^2 - 4t$ Clarissa N.

### Problem 8

Differentiate the function.
$f(t) = 1.4t^5 - 2.5t^2 + 6.7$ Clarissa N.

### Problem 9

Differentiate the function.
$g(x) = x^2(1-2x)$ Clarissa N.

### Problem 10

Differentiate the function.
$H(u) = (3u - 1)(u + 2)$ Clarissa N.

### Problem 11

Differentiate the function.
$g(t) = 2t^{-3/4}$ Clarissa N.

### Problem 12

Differentiate the function.
$B(y) = cy^{-6}$ Clarissa N.

### Problem 13

Differentiate the function.
$F(r) = \frac{5}{r^3}$ Clarissa N.

### Problem 14

Differentiate the function.
$y = x^{5/3} - x^{2/3}$ Clarissa N.

### Problem 15

Differentiate the function.
$R(a) = (3a + 1)^2$ Clarissa N.

### Problem 16

Differentiate the function.
$h(t) = \sqrt{t} - 4e^1$ Clarissa N.

### Problem 17

Differentiate the function.
$S(p) = \sqrt{p} - p$ Clarissa N.

### Problem 18

Differentiate the function.
$y = \sqrt{x}(2 + x )$ Clarissa N.

### Problem 19

Differentiate the function.
$y = 3e^x + \frac{4}{\sqrt{x}}$ Clarissa N.

### Problem 20

Differentiate the function.
$S(R) = 4\pi R^2$ Clarissa N.

### Problem 21

Differentiate the function.
$h(u) = Au^3 + Bu^2 + Cu$ Clarissa N.

### Problem 22

Differentiate the function.
$y = \frac {\sqrt{x + x}}{x^2}$ Clarissa N.

### Problem 23

Differentiate the function.
$y = \frac {x^2+4x+3}{\sqrt{x}}$ Clarissa N.

### Problem 24

Differentiate the function.
$G(t) = \sqrt{5t} + \frac {\sqrt{7}}{t}$ Clarissa N.

### Problem 25

Differentiate the function.
$j(x) = x^{2.4} + e^{2.4}$ Clarissa N.

### Problem 26

Differentiate the function.
$k(r) = e^r + r^e$ Clarissa N.

### Problem 27

Differentiate the function.
$G(q) = (1 + q^{-1})^2$ Clarissa N.

### Problem 28

Differentiate the function.
$F(z) = \frac{A + Bz + Cz^2}{z^2}$ Clarissa N.

### Problem 29

Differentiate the function.
$f(v) = \frac{\sqrt{v - 2ve^v}}{v}$ Clarissa N.

### Problem 30

Differentiate the function.
$D(t) = \frac {1 + 16t^2}{(4t)^3}$ Clarissa N.

### Problem 31

Differentiate the function.
$z = \frac {A}{y^{10}} + Be^y$ Carson M.

### Problem 32

Differentiate the function.
$y = e^{x+1} + 1$ Clarissa N.

### Problem 33

Find an equation of the tangent line to the curve at the given point.
$y = 2x^3 - x^2 + 2, (1,3)$ Clarissa N.

### Problem 34

Find an equation of the tangent line to the curve at the given point.
$y = 2e^x + x, (0,2)$ Clarissa N.

### Problem 35

Find an equation of the tangent line to the curve at the given point.
$y = x + \frac{2}{x} , (2,3)$ Clarissa N.

### Problem 36

Find an equation of the tangent line to the curve at the given point.
$y = \sqrt{x} - x, (1,0)$ Clarissa N.

### Problem 37

Find equations of the tangent line and normal line to the curve at the given point.
$y = x^4 + 2e^x, (0,2)$ Clarissa N.

### Problem 38

Find equations of the tangent line and normal line to the curve at the given point.
$y^2 = x^3, (1,1)$ Clarissa N.

### Problem 39

Fidn an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same scree.
$y = 3x^2 - x^3, (1,2)$ Clarissa N.

### Problem 40

Fidn an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same scree.
$y = x - \sqrt{x}, (1,0)$ Clarissa N.

### Problem 41

Find $f'(x)$. Compare the graphs of $f$ and $f'$ and use them to explain why your answer is reasonable.
$f(x) = x^4 - 2x^3 + x^2$ Clarissa N.

### Problem 42

Find $f'(x)$. Compare the graphs of $f$ and $f'$ and use them to explain why your answer is reasonable.
$f(x) = x^5 - 2x^3 + x - 1$ Clarissa N.

### Problem 43

(a) Graph the function
$f(x) = x^4 - 3x^3 - 6x^2 + 7x + 30$
in the viewing rectangle [-3,5] by [-10,50].
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $f'$.
(c) Calculate $f'(x)$ and use this expression, with graphing device, to graph $f'$. Compare with your sketch in part (b). Clarissa N.

### Problem 44

(a) Graph the function $g(x) = e^x - 3x^2$ in the viewing rectangle [-1,4] by [-8,8].
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $g'$.
(c) Calculate $g'(x)$ and use this expression, with a graphing device, to graph $g'$. Compare with your sketch in part (b). Clarissa N.

### Problem 45

Find the first and second derivatives of the function.
$f(x) = 0.001x^5 - 0.02x^3$ Clarissa N.

### Problem 46

Find the first and second derivatives of the function.
$G(r) = \sqrt{r} + \sqrt{r}$ Clarissa N.

### Problem 47

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f, f',$ and $f".$
$f(x) = 2x - 5x^{3/4}$ Clarissa N.

### Problem 48

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f, f',$ and $f".$
$f(x) = e^x - x^3$ Clarissa N.

### Problem 49

The equation of motion of a particle is $= t^3 - 3t,$ where is in meters and is in seconds. Find
(a) the velocity and acceleration as functions of $t,$
(b) the acceleration after $2 s,$ and
(c) the acceleration when the velocity is 0. Clarissa N.

### Problem 50

The equation of motion of a particle is, $s = 2t^4 - 2t^3 + t^2 - t$ , where is in meters and is in
seconds.
(a) Find the velocity and acceleration as functions of .
(b) Find the acceleration after 1 s.
(c) Graph the position, velocity, and acceleration functions on the same screen. Clarissa N.

### Problem 51

Biologists have proposed a cubic polynomial to model the length $L$ of Alaskan rockfish at age $A:$
$L = 0.0155A^3 - 0.372A^2 + 3.95A + 1.21$
where $L$ is measured in inches and $A$ in years. Calculate
$\frac{dL}{dA}\mid A=12$ Clarissa N.

### Problem 52

The number of tree species $S$ in a given area $A$ in the Pasoh Forest Reserve in Malaysia has been modeled by the power function

$S(A) = 0.882A^0.842$

where A is measured in square meters. Find $S' (100)$ and interpret your answer. Clarissa N.

### Problem 53

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure $P$ of the gas is inversely proportional to the volume $V$ of the gas.
(a) Suppose that the pressure of a sample of air that occupies $0.106 m^3 at 25^{\circ} C$ is 50 kPa. Write $V$ as a function of $P.$
(b) Calculate $dV\mid dP$ when $P$ = 50 kPA. What is the meaning of the derivative? What are its units? Clarissa N.

### Problem 54

Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in tge tabe shoe tire life $L$ (in thousands of miles) for a certain type of tire at various pressures $P (in lb/in^2).$

(a) Use a calculator to model tire life with a quadratic function of the pressure.
(b) Use the model to estimate $dL/dP$ when $P = 30$ and when $P = 40.$ What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives? Clarissa N.

### Problem 55

Find the points on the curve $y = 2x^3 + 3x^2 - 12x + 1$ where the tangent is horizontal. Clarissa N.

### Problem 56

For what value of $x$ does the graph of $f(x) = e^x - 2x$ have a horizontal tangent? Clarissa N.

### Problem 57

Show that the curve $y = 2e^x + 3x + 5x^3$ has no tangent line with slope 2. Clarissa N.

### Problem 58

Find an equation of the tangent line to the curve $y = x^4 + 1$ that is parallel to the line $32^x - y = 15.$ Clarissa N.

### Problem 59

Find equation of both lines that are tangent to the curve $y = x^3 - 3x^2 + 3x - 3$ and are parallel to the line $3x - y = 15.$ Clarissa N.

### Problem 60

At what point on the curve $y = 1 + 2e^x - 3x$ is the tangent line parallel to the line $3x - y = 5?$ Illustrate with a sketch. Clarissa N.

### Problem 61

Find an equation of the normal line to the curve $y = \sqrt{x}$ that is parallel to the line $2x + y = 1.$ Clarissa N.

### Problem 62

Where does the normal line to the parabola $y = x^2 - 1$ at the point (-1, 0) intersect the parabola a second time? Illustrate with a sketch. Clarissa N.

### Problem 63

Draw a diagram to shoe that there are two tangent lines to the parabola $y = x^2$ that pass through the point (0, -4). Find the coordinates of the point where these tangent lines intersect the parabola. Clarissa N.

### Problem 64

(a) Find equations of both lines through the point (2, -3) that are tangent to the parabola $y = x^2 + x.$
(b) Show that there is no line through the point (2,7) that is tangent to the parabola. Then draw a diagram to see why. Clarissa N.

### Problem 65

Use the definition of a derivative to show that if $f(x) = 1/x,$ then $f'(x) = -1/x^2.$ (This proves the Power Rule for the case $n = -1.$ ) Clarissa N.

### Problem 66

Find the $n$ th derivative of each function by calculating the first few derivatives and observing the pattern that occurs.
(a) $f(x) = x"$ (b) $f(x) = 1/x$ Clarissa N.

### Problem 67

Find a second-degree polynomial & P & such that $P(2) = 5, P'(2) = 3,$ and $P"(2) = 2.$ Clarissa N.

### Problem 68

The equation $y" + y' - 2y = x^2$ is called a differential equation because it involves an unknown function $y$ and its derivatives $y'$ and $y".$ Find constant $A, B,$ and $C$ such that the function $y = Ax^2 + Bx + C$ satisfies this equation. (Differential equations will be studied in detail in Chapter 9.) Clarissa N.

### Problem 69

Find a cubic function $y = ax^3 + bx^2 + cx + d$ whose graph has horizontal tangent at the points (-2, 6) and (2, 0). Clarissa N.

### Problem 70

Find a parabola with equation $y = ax^2 + bx + c$ that has slope 4 at $x = 1,$ slope -8 at $x = -1,$ and passes through the point (2, 15). Clarissa N.

### Problem 71

Let.
$f(x) = \left\{ \begin{array}{ll} x^2 + 1 & \mbox{if} x < 1\\ x + 1 & \mbox{if} x \ge 1 \\ \end{array} \right.$
Is $f$ differentiable at 1? Sketch the graphs of $f$ and $f '.$

FL
Frank L.

### Problem 72

At what numbers is the following function $g$ differentiable?
$g(x) = \left\{ \begin{array}{ll} 2x & \mbox{if}x \le 0\\ 2x - x^2 & \mbox{if}0 < x < 2\\ 2 - x & \mbox{if}x \ge 2 \end{array} \right.$
Give a formula for $g'$ and sketch the graphs of $g$ and $g'.$

FL
Frank L.

### Problem 73

(a) For what values of $x$ is the function $f(x) = \mid x^2 - 9 \mid$ differentiable? Find a formula for $f'.$
(b) Sketch the graph of $f$ and $f'.$ Clarissa N.

### Problem 74

Where is the function $h(x) = \mid x - 1 \mid + \mid x + 2 \mid$ differentiable? Give a formula for $h'$ and sketch the graph of $h$ and $h'.$ Clarissa N.

### Problem 75

Find the parabola with equation $y = ax^2 + bx$ whose tangent line at (1, 1) has equation $y = 3x - 2.$ Clarissa N.

### Problem 76

Suppose the curve $y = x^4 + ax^3 + bx^2 + cx + d$ has a tangent line when $x = 0$ with equation $y = 2x + 1$ and tangent line when $x = 1$ with equation $y = 2 - 3x.$ Find the values of $a,b,c,$ and $d.$ Clarissa N.

### Problem 77

For what values of $a$ and $b$ is the line $2x + y = b$ tangent to the parabola $y = ax^2$ when $x = 2?$ Clarissa N.
\end{array} \right. $Find the values of$ m $and$ b $that make$ f $differentiable everywhere. Clarissa N. Numerade Educator ### Problem 82 A tangent line is drawn to the hyperbola$ xy = c $at a point$ P. $(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is$ P. $(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where$ P $is located on the hyperbola. Clarissa N. Numerade Educator ### Problem 83 Evaluate$ \lim_{x\to1} \frac {x^{1000} - 1}{x - 1}. $ Clarissa N. Numerade Educator ### Problem 84 Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola$ y = x^2. $Where do these lines intersect? Clarissa N. Numerade Educator ### Problem 85 If$ c > \frac {1}{2}, $how many lines through the point (0, c) are normal lines to the parabola$ y = x^2? $What if$ c \le \frac {1}{2}? $ Clarissa N. Numerade Educator ### Problem 86 Sketch the parabolas$ y = x^2 and $y = x^2 - 2x + 2.$ Do you think there is a line that is tangent to both curves? If so. find its equation. If not, why not? 