{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?

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 Sup​​pose 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 Sup​​pose 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} 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 jumpe​​r 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) = 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)

{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.

y = 8/(x² + 4) (Witch of Agnesi)

Sketch the graph of a function continuous on the given interval that satisfies the following conditions.

ƒ is continuous on the interval [-4, 4] ; f'(x) = 0 for x = -2, 0, and 3; ƒ has an absolute minimum at x = 3; ƒ has a local minimum at x = -2 ; ƒ has a local maximum at x = 0; ƒ has an absolute maximum at x = -4.

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

Plot a possible graph of f.

Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

ƒ(x) = (x⁴/2) - 3x² + 4x + 1

Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

ƒ(x) = 3x/(x² + 3)

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

f(x) = 3x/(x² - 1)

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

f(x) = x²/(x - 2)

{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.

x⁴ - x² + y² = 0 (Figure-8 curve)

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.

a. Plot a graph of the curve when a = 3.

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.

a. Determine the marginal cost and average cost functions. Graph and interpret these functions.

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

f(x) = (x² + 12)/(2x + 1)

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

f(x) = tan⁻¹ (x²/√3)

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

f(x) = x⁴ + 4x³

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

f(x) = x³ - 6x² - 135x

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

ƒ(x) = 4cos (π (x-1)) on [0, 2]

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ

{Use of Tech} A pursuit curve A man stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr . The path in the xy-plane followed by the man as he pursues the dog is given by the function y = ƒ(x) = s/2 ((x(ˢ⁺¹)/ˢ) /(s+1) - (x(ˢ⁺¹)/ˢ / s-1)) + s/ s² - 1

Select various values of s > 1 and graph this pursuit curve. Comment on the changes in the curve as s increases. <IMAGE>

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.

c. By experimentation, determine the approximate value of a (3 < a < 4)at which the graph separates into two curves.