Calculus: Early Transcendentals

(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 $?

(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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Differentiate the function.
$ y = \sqrt[3]{x}(2 + x ) $

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find an equation of the tangent line to the curve at the given point.
$ y = \sqrt[4]{x} - x, (1,0) $

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

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

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) $

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) $

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 $

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 $

(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).

(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).

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

Find the first and second derivatives of the function.
$ G(r) = \sqrt{r} + \sqrt[3]{r} $

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} $

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 $

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.

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.

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 $

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.

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?

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?

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

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

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

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

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. $

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.

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

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.

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.

(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.

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.$ )

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 $

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

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.)

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

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).

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 '. $

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'. $

(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'. $

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'. $

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

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. $

For what values of $ a $ and $ b $ is the line $ 2x + y = b $ tangent to the parabola $ y = ax^2 $ when $ x = 2? $

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

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

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.

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.

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.

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

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?

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}? $

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?