Problem 6: The graph of a function f(x) is shown below.

a) Check the answers in the table below and correct those that are erroneous.

Roughly, f '(1) and f '(5) are as stated in the table, but f '(2) = -2, f '(3) = 3 and f '(4) = 2. Here is a plot of f '(x):

Problem 7: The position of a moving object (measured in feet) is given by p(t) = -2t^3 + 15t^2-22t, where t represents time measured in minutes.

a) Find a formula for the velocity of the object at time t.

b) Is the object moving forwards or backwards at t = 1? How about t = 3?

c) Find the time(s) when the objects velocity is -8.5 ft/min.

The velocity is the derivative of the position function:

> vel:=diff(-2*t^3+15*t^2-22*t,t);

Positive velocity means forward and negative velocity means backwards. Just substitute t = 1 and t = 3 into the velocity formula.

> at_one = subs({t=1},vel);
at_three = subs({t=3},vel);

A plot of y = -8.5 and the velocity:

> plot([vel,-8.5],t=0..5,color=[red,blue]);

Using your calculator, or the quadratic formula, you should be able to locate those 2 intersection points (there are no others). Here are the t values you should find.

> times =solve(vel=-8.5,t);

Problem 8: An economist is interested in how the price of a certain commodity affects its sales. Suppose that at a price of $p, a quantity, q, of the commodity is sold. If q = f(p), explain in economic terms the meaning of the statements f(10) = 240000 and f '(10) = -21000. Would you suggest a price greater than $10 or less than $10? Why?

At a selling price of $10 each, 240000 units will be sold, and at a price of $10 each, each dollar increase in price will generate the sales of 21000 fewer units (however, this rate of change is instantaneous and may vary once the price changes). Total income at $10 is 10*240000 = $2400000. Assuming the rate of change stays approximately at -21000, at a price of $11, 240000 - 21000 = 219000 units will be sold, giving 11*219000 = $2409000 in sales. I'd suggest a price higher than $10 based on this. Likewise, if the price was lowered to $9, 240000 + 21000 (approximately) units would be sold, yielding 261000*9 = $2349000 in sales. Again a price higher than $10 seems better.

Problem 9: The graph of a function f(x) is shown below. Use the graph to answer the following questions.

a) For which values of x does f '(x) = 0?

Approximately x = 0.7 and 3.8

b) Is f '(x) positive or negative at x = 2? How can you tell?

Positive since f(x) is increasing near x = 2.

c) Is f '(x) increasing or decreasing at x = 3.5? How do you know?

Decreasing, the slopes are getting less positive (f(x) is concave down near x = 3.5).

d) Is f ''(1) positive or negative? How do you know?

Positive, since f(x) is concave up at x = 1 (the slopes are getting steeper).

e) Somewhere between x = 1.5 and x = 3.5, f '(x) has a hilltop (local maximum). Estimate the x-coordinate of this point and the value of f '(x) at this point.

The inflection point (where f(x) has its steepest uphill slope) seems to be around x = 2.5. Drawing in the tangent there gives a slope close to 3.4

Problem 10: The graph of the derivative, f '(x) is shown below. The function f(x) is not shown, yet all questions can answered from just the graph of the derivative.

a) Is f(x) increasing or decreasing at x = 0? How can you tell?

The derivative is positive at x = 0, so f(x) is increasing at x = 0.

b) Is f(x) concave up or concave down at x = 2? How can you tell?

At x = 2, f '(x) is decreasing (so f ''(x) is negative), so f(x) is concave down.

c) If f(1) = 2, find the equation of the tangent line to f(x) at x = 1.

The slope is f '(1) = -0.75 (reading the graphs y-coordinate at x = 1) and (1, f(1) ) = (1, 2) is on the tangent line, therefore the equation is y = -0.75(x-1) + 2.

d) Which value(s) of x (between 0 and 5) are local minimums (valley bottoms) on the graph of f(x)? How about hilltop(s) (local maximums)? What about inflection point(s)?

x = 4.1 is a local minimum of f(x), as to the left of x = 4.1, f(x) is decreasing, while to the right of x = 4.1, f(x) is increasing. The tangent line at x = 4.1 has a slope of zero. Similarly, x = 0.6 is a hilltop (f(x) is increasing, flat then decreasing near x = 0.6). That's it for hilltops and valley bottoms. The valley bottom of f '(x), at x = 2.6, is where f(x) switches from (decreasing and) concave down to (decreasing and) concave up, i.e. x = 2.6 is an inflection point of f(x).

e) If f(0) = 2, sketch a rough graph of f(x). Your graph should look something like the blue graph below:

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