5. Graphical Applications of Derivatives

Let ƒ(x) = (x - 3) (x + 3)²

f. State the x- and y-intercepts of the graph of ƒ.

Mean Value Theorem and graphs Find all points on the interval (1,3) at which the slope of the tangent line equals the average rate of change of f on [1,3]. Reconcile your results with the Mean Value Theorem. <IMAGE>

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.

a. Graph and interpret the mileage function.

{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.

b. Graph and interpret the gas mileage m(g)/g. 

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

f(x) = x² - 6x

Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.

f"(x) > 0 on (-∞,-2); f"(-2) = 0; f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4,∞)

Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.

f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>

f. Sketch one possible graph of f.

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < 3

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < -1

Sketch a continuous function f on some interval that has the properties described. Answers will vary.

The function f satisfies f'(-2) = 2, f'(0) = 0, f'(1) = -3 and f'(4) = 1.

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for 8 < x < 10

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?