(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

(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

Differentiate the function.

$ g(x) = \frac{7}{4}x^2 - 3x + 12 $

Clarissa N.

Numerade Educator

Differentiate the function.

$ y = 3e^x + \frac{4}{\sqrt[3]{x}} $

Clarissa N.

Numerade Educator

Differentiate the function.

$ y = \frac {\sqrt{x + x}}{x^2} $

Clarissa N.

Numerade Educator

Differentiate the function.

$ y = \frac {x^2+4x+3}{\sqrt{x}} $

Clarissa N.

Numerade Educator

Differentiate the function.

$ G(t) = \sqrt{5t} + \frac {\sqrt{7}}{t} $

Clarissa N.

Numerade Educator

Differentiate the function.

$ F(z) = \frac{A + Bz + Cz^2}{z^2} $

Clarissa N.

Numerade Educator

Differentiate the function.

$ f(v) = \frac{\sqrt[3]{v - 2ve^v}}{v} $

Clarissa N.

Numerade Educator

Differentiate the function.

$ D(t) = \frac {1 + 16t^2}{(4t)^3} $

Clarissa N.

Numerade Educator

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

Find an equation of the tangent line to the curve at the given point.

$ y = 2e^x + x, (0,2) $

Clarissa N.

Numerade Educator

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

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

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

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

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

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

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

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

(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

(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

Find the first and second derivatives of the function.

$ f(x) = 0.001x^5 - 0.02x^3 $

Clarissa N.

Numerade Educator

Find the first and second derivatives of the function.

$ G(r) = \sqrt{r} + \sqrt[3]{r} $

Clarissa N.

Numerade Educator

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

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

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

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

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

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

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

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

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

Clarissa N.

Numerade Educator

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

Clarissa N.

Numerade Educator

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

Clarissa N.

Numerade Educator

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

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

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

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

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

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

(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

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

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

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

Clarissa N.

Numerade Educator

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

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

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

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 '. $

Frank L.

Numerade Educator

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'. $

Frank L.

Numerade Educator

(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.

Numerade Educator

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.

Numerade Educator

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

Clarissa N.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

Find the value of $ c $ such that the line $ y = \frac{3}{2}x + 6 $ is tangent to the curve $ y = c \sqrt{x}.

Clarissa N.

Numerade Educator

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

Clarissa N.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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

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

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

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?

Clarissa N.

Numerade Educator