5. Graphical Applications of Derivatives

Local max/min of x¹⸍ˣ Use analytical methods to find all local extrema of the function ƒ(x) = x¹⸍ˣ , for x > 0 . Verify your work using a graphing utility.

{Use of Tech} Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W=1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh=3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t+4t²−t³ / 9kWh where t=0 corresponds to midnight.

c. Graph the power function and interpret the graph. What are the units of power in this case?

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

a. Graph the volume function. What is the volume of water in the tank before the valve is opened? 

{Use of Tech} Bungee jumper A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^−t cos t), for t ≥ 0.

b. Use a graphing utility to determine when she is moving downward and when she is moving upward during the first 10 s.  

Let ƒ(x) = (x - 3) (x + 3)²

g. Use your work in parts (a) through (f) to sketch a graph of ƒ.

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

g. Use your work in parts (a) through (f) to sketch a graph of ƒ .

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x³ - 6x² + 9x

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = ln (x² + 1)

Use the guidelines of this section to make a complete graph of f.

f(x) = x + 2 cos x on [-2π,2π)

Use the guidelines of this section to make a complete graph of f.

f(x) = 2 - 2x^2/3 + x^4/3

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = 1/(e⁻ˣ - 1)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = e⁻ˣ²/₂

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.

ƒ(x) = 1/3 x³ - 2x² - 5x + 2

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x³ - 147x + 286

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)