Ask your homework questions to teachers and professors, meet other students, and be entered to win $600 or an Xbox Series X 🎉Join our Discord! ## Chapter 3 ## Differentiation Rules ## Educators AL WZ + 4 more educators ### Problem 1 (a) How is the number$ 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 $? Clarissa N. Numerade Educator ### Problem 2 (a) Sketch, by hand, the graph of the function$ 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 $is large? Clarissa N. Numerade Educator ### Problem 3 Differentiate the function.$ f(x) = 2^{40} $Clarissa N. Numerade Educator ### Problem 4 Differentiate the function.$ f(x) = e^5 $Clarissa N. Numerade Educator ### Problem 5 Differentiate the function.$ f(x) = 5.2x + 2.3 $Clarissa N. Numerade Educator ### Problem 6 Differentiate the function.$ g(x) = \frac{7}{4}x^2 - 3x + 12 $Clarissa N. Numerade Educator ### Problem 7 Differentiate the function.$ f(t) = 2t^3 - 3t^2 - 4t $Clarissa N. Numerade Educator ### Problem 8 Differentiate the function.$ f(t) = 1.4t^5 - 2.5t^2 + 6.7 $Clarissa N. Numerade Educator ### Problem 9 Differentiate the function.$ g(x) = x^2(1-2x) $Clarissa N. Numerade Educator ### Problem 10 Differentiate the function.$ H(u) = (3u - 1)(u + 2) $Clarissa N. Numerade Educator ### Problem 11 Differentiate the function.$ g(t) = 2t^{-3/4} $Clarissa N. Numerade Educator ### Problem 12 Differentiate the function.$ B(y) = cy^{-6} $Clarissa N. Numerade Educator ### Problem 13 Differentiate the function.$ F(r) = \frac{5}{r^3} $Clarissa N. Numerade Educator ### Problem 14 Differentiate the function.$ y = x^{5/3} - x^{2/3} $Clarissa N. Numerade Educator ### Problem 15 Differentiate the function.$ R(a) = (3a + 1)^2 $Clarissa N. Numerade Educator ### Problem 16 Differentiate the function.$ h(t) = \sqrt[4]{t} - 4e^1 $Clarissa N. Numerade Educator ### Problem 17 Differentiate the function.$ S(p) = \sqrt{p} - p $Clarissa N. Numerade Educator ### Problem 18 Differentiate the function.$ y = \sqrt[3]{x}(2 + x ) $Clarissa N. Numerade Educator ### Problem 19 Differentiate the function.$ y = 3e^x + \frac{4}{\sqrt[3]{x}} $Clarissa N. Numerade Educator ### Problem 20 Differentiate the function.$ S(R) = 4\pi R^2 $Clarissa N. Numerade Educator ### Problem 21 Differentiate the function.$ h(u) = Au^3 + Bu^2 + Cu $Clarissa N. Numerade Educator ### Problem 22 Differentiate the function.$ y = \frac {\sqrt{x + x}}{x^2} $Clarissa N. Numerade Educator ### Problem 23 Differentiate the function.$ y = \frac {x^2+4x+3}{\sqrt{x}} $Clarissa N. Numerade Educator ### Problem 24 Differentiate the function.$ G(t) = \sqrt{5t} + \frac {\sqrt{7}}{t} $Clarissa N. Numerade Educator ### Problem 25 Differentiate the function.$ j(x) = x^{2.4} + e^{2.4} $Clarissa N. Numerade Educator ### Problem 26 Differentiate the function.$ k(r) = e^r + r^e $Clarissa N. Numerade Educator ### Problem 27 Differentiate the function.$ G(q) = (1 + q^{-1})^2 $Clarissa N. Numerade Educator ### Problem 28 Differentiate the function.$ F(z) = \frac{A + Bz + Cz^2}{z^2} $Clarissa N. Numerade Educator ### Problem 29 Differentiate the function.$ f(v) = \frac{\sqrt[3]{v - 2ve^v}}{v} $Clarissa N. Numerade Educator ### Problem 30 Differentiate the function.$ D(t) = \frac {1 + 16t^2}{(4t)^3} $Clarissa N. Numerade Educator ### Problem 31 Differentiate the function.$ z = \frac {A}{y^{10}} + Be^y $Carson M. Numerade Educator ### Problem 32 Differentiate the function.$ y = e^{x+1} + 1 $Clarissa N. Numerade Educator ### Problem 33 Find an equation of the tangent line to the curve at the given point.$ y = 2x^3 - x^2 + 2, (1,3) $Clarissa N. Numerade Educator ### Problem 34 Find an equation of the tangent line to the curve at the given point.$ y = 2e^x + x, (0,2) $Clarissa N. Numerade Educator ### Problem 35 Find an equation of the tangent line to the curve at the given point.$ y = x + \frac{2}{x} , (2,3) $Clarissa N. Numerade Educator ### Problem 36 Find an equation of the tangent line to the curve at the given point.$ y = \sqrt[4]{x} - x, (1,0) $Clarissa N. Numerade Educator ### Problem 37 Find equations of the tangent line and normal line to the curve at the given point.$ y = x^4 + 2e^x, (0,2) $Clarissa N. Numerade Educator ### Problem 38 Find equations of the tangent line and normal line to the curve at the given point.$ y^2 = x^3, (1,1) $Clarissa N. Numerade Educator ### Problem 39 Fidn an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same scree.$ y = 3x^2 - x^3, (1,2) $Clarissa N. Numerade Educator ### Problem 40 Fidn an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same scree.$ y = x - \sqrt{x}, (1,0) $Clarissa N. Numerade Educator ### Problem 41 Find$ 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 $Clarissa N. Numerade Educator ### Problem 42 Find$ 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 $Clarissa N. Numerade Educator ### Problem 43 (a) Graph the function$ 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' $. Compare with your sketch in part (b). Clarissa N. Numerade Educator ### Problem 44 (a) Graph the function$ 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' $. Compare with your sketch in part (b). Clarissa N. Numerade Educator ### Problem 45 Find the first and second derivatives of the function.$ f(x) = 0.001x^5 - 0.02x^3 $Clarissa N. Numerade Educator ### Problem 46 Find the first and second derivatives of the function.$ G(r) = \sqrt{r} + \sqrt[3]{r} $Clarissa N. Numerade Educator ### Problem 47 Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of$ f, f', $and$ f".  f(x) = 2x - 5x^{3/4} $Clarissa N. Numerade Educator ### Problem 48 Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of$ f, f', $and$ f".  f(x) = e^x - x^3 $Clarissa N. Numerade Educator ### Problem 49 The equation of motion of a particle is$ = 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, $and (c) the acceleration when the velocity is 0. Clarissa N. Numerade Educator ### Problem 50 The equation of motion of a particle is,$ s = 2t^4 - 2t^3 + t^2 - t $, where is in meters and is in seconds. (a) Find the velocity and acceleration as functions of . (b) Find the acceleration after 1 s. (c) Graph the position, velocity, and acceleration functions on the same screen. Clarissa N. Numerade Educator ### Problem 51 Biologists have proposed a cubic polynomial to model the length$ 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 $Clarissa N. Numerade Educator ### Problem 52 The number of tree species$ 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) $and interpret your answer. Clarissa N. Numerade Educator ### Problem 53 Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure$ 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 $= 50 kPA. What is the meaning of the derivative? What are its units? Clarissa N. Numerade Educator ### Problem 54 Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in tge tabe shoe tire life$ 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. $What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives? Clarissa N. Numerade Educator ### Problem 55 Find the points on the curve$ y = 2x^3 + 3x^2 - 12x + 1 $where the tangent is horizontal. Clarissa N. Numerade Educator ### Problem 56 For what value of$ x $does the graph of$ f(x) = e^x - 2x $have a horizontal tangent? Clarissa N. Numerade Educator ### Problem 57 Show that the curve$ y = 2e^x + 3x + 5x^3 $has no tangent line with slope 2. Clarissa N. Numerade Educator ### Problem 58 Find an equation of the tangent line to the curve$ y = x^4 + 1 $that is parallel to the line$ 32^x - y = 15. $Clarissa N. Numerade Educator ### Problem 59 Find equation of both lines that are tangent to the curve$ y = x^3 - 3x^2 + 3x - 3 $and are parallel to the line$ 3x - y = 15. $Clarissa N. Numerade Educator ### Problem 60 At what point on the curve$ y = 1 + 2e^x - 3x $is the tangent line parallel to the line$ 3x - y = 5? $Illustrate with a sketch. Clarissa N. Numerade Educator ### Problem 61 Find an equation of the normal line to the curve$ y = \sqrt{x} $that is parallel to the line$ 2x + y = 1. $Clarissa N. Numerade Educator ### Problem 62 Where does the normal line to the parabola$ y = x^2 - 1 $at the point (-1, 0) intersect the parabola a second time? Illustrate with a sketch. Clarissa N. Numerade Educator ### Problem 63 Draw a diagram to shoe that there are two tangent lines to the parabola$ y = x^2 $that pass through the point (0, -4). Find the coordinates of the point where these tangent lines intersect the parabola. Clarissa N. Numerade Educator ### Problem 64 (a) Find equations of both lines through the point (2, -3) that are tangent to the parabola$ y = x^2 + x. $(b) Show that there is no line through the point (2,7) that is tangent to the parabola. Then draw a diagram to see why. Clarissa N. Numerade Educator ### Problem 65 Use the definition of a derivative to show that if$ f(x) = 1/x, $then$ f'(x) = -1/x^2. $(This proves the Power Rule for the case$ n = -1.$) Clarissa N. Numerade Educator ### Problem 66 Find the$ 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 $Clarissa N. Numerade Educator ### Problem 67 Find a second-degree polynomial & P & such that$ P(2) = 5, P'(2) = 3, $and$ P"(2) = 2. $Clarissa N. Numerade Educator ### Problem 68 The equation$ 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 $satisfies this equation. (Differential equations will be studied in detail in Chapter 9.) Clarissa N. Numerade Educator ### Problem 69 Find a cubic function$ y = ax^3 + bx^2 + cx + d $whose graph has horizontal tangent at the points (-2, 6) and (2, 0). Clarissa N. Numerade Educator ### Problem 70 Find a parabola with equation$ y = ax^2 + bx + c $that has slope 4 at$ x = 1, $slope -8 at$ x = -1, $and passes through the point (2, 15). Clarissa N. Numerade Educator ### Problem 71 Let.$ 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 '. $FL Frank L. Numerade Educator ### Problem 72 At what numbers is the following function$ 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

Clarissa N.

### Problem 79

What is the value of $c$ such that the line $y = 2x + 3$ is tangent to the parabola $y = cx^2?$

Clarissa N.

### Problem 80

The graph of any quadratic function $f(x) = ax^2 + bx + c$ is a parabola. Prove that the average of the slopes of the slopes of the tangent lines to the parabola at the endpoints of any interval $[p, q]$ equals the slope of the tangent line at the midpoint of the interval.

Clarissa N.

### Problem 81

Let
$f(x) = \left\{ \begin{array}{ll} x^2 + & \mbox{if} x \le 2\\ mx + b & \mbox{if} x > 2\\ \end{array} \right.$
Find the values of $m$ and $b$ that make $f$ differentiable everywhere.

Clarissa N.

### 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.

### Problem 83

Evaluate $\lim_{x\to1} \frac {x^{1000} - 1}{x - 1}.$

Clarissa N.

### 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.

### 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.
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?