(2) Let f(x) = 5x. (3) Say f (x) and f (x) have the following graphs. Sketch a graph of f(x). The graph of f (x) is: 3x 5

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1 The following review sheet is intended to help you study. It does not contain every type of problem you may see. It does not reflect the distribution of problems on the actual midterm. It probably has mistakes (if you find one, tell us!). But, it can give you some extra practice with most of the main concepts covered in the course. Derivatives, Limits, and Graphing The beginning of Chapter, from. to.4, is almost like one large section all about how derivatives and limits can be used to find information about graphs and vice versa. You should be well-versed in the relationship between functions, derivatives, second derivatives, relative extrema, increasing and decreasing, and concavity. You should know how to find end behavior and asymptotes when they exist by using limits. You should be able to put all this together to build a graph of a function when you re given its formula. () Let f(x) = x 3 + 3x 2 9x +. Find critical points of f, find the intervals where f is increasing and decreasing, and any relative extrema of f. Find all inflection values, and intervals where f is concave up and concave down. Find if f has any asymptotes. Graph f. (2) Let f(x) = x 3x. Find critical points of f, find the intervals where f is increasing and decreasing, and any relative extrema of f. Find all inflection values, and intervals where f is concave up and concave down. Find if f has any asymptotes. Graph f. (3) Say f (x) and f (x) have the following graphs. Sketch a graph of f(x). The graph of f (x) is: The graph of f (x) is:

2 (4) Let s say the graph of f(x) is given below. Sketch a graph of f (x). It s a little harder, but also try to sketch a graph of f (x). The graph of f(x) is: () Evaluate the following limits. (a) lim x x x 2 (b) lim t 3 7 t 2 9 (c) lim x 4 x 2 6 (d) lim x (e) lim x (f) lim x 3x 4 x+ 7x 4 +2x 3 8 3x 4 x+ 7x +2x 3 8 3x 4 x+ 7x 3 +2x 3 8 (g) lim x e x e x (h) lim x e x e x (i) lim x e x2 e x + (6) Find which of the following functions have vertical asymptotes. If a function does have a vertical asymptote, find where it is, and whether the function approaches positive or negative infinity on each side of the asymptote. (a) f(x) = x+2 x+3 (b) g(z) = z2 z 2 + (c) h(q) = q2 + q 2 (d) m(y) = y4 3y 2 +2

3 Absolute Extrema A major application of what we know about derivatives is in finding absolute extrema. Use what you know about absolute extrema and derivatives to solve the following. (7) Find the absolute extrema of f(x) = x 3 3x on the following intervals. (a) [, 2] (b) [0, ) (c) [, 3] (8) Find the absolute extrema of f(x) = x 3 9x x + on the interval [0, ]. (9) Find all critical points of the function f(x) = ln x x. Decide whether each is a local max, local min, or neither. Find the absolute maximum and absolute minimum of f on the interval [, e 2 ]. Optimization Of course, we usually have a reason for wanting to find a maximum or a minimum. These reasons come in the form of optimization problems. Yes, the problems you get in this class are simplified, somewhat contrived examples, but this really is useful stuff! (0) Suppose Farmer Fred has 800 yards of fencing. Farmer Fred s land has a river running through it, and he wishes to fence in a pasture using a rectangular shape where one side is formed by the river (assuming the river is straight). What are the dimensions of the largest pasture that Farmer Fred can fence? () Blaney the blogger has a website, which consists of two parts one part is accessible for free, and the other part costs $0 per month to access. Blaney currently has 00 subscribers, and she knows that for every dollar she raises the price, she will lose 20 subscribers (if she lowers the price below $0, nothing will happen). More subscribers means more maintenance, though it costs Blaney $.20 per month for each subscriber that she has. (a) Let x = Blaney s monthly access fee, and let y = the number of subscribers Blaney has. Write an equation expressing Blaney s revenue per month as a function of x and y. (b) Write an equation relating x and y, and solve it for y (so that y is expressed in terms of x). (c) Write a function for Blaney s cost per month as a function of x. Write a function for Blaney s profit per month as a function of x. (d) What monthly access fee will maximize Blaney s profit? At that price, how much profit will she earn per month? (2) Old Maconald has a farm, and various animals that he needs to build an enclosure for. He will build a square enclosure for his cats and a circular enclosure for his dolphins. The square enclosure costs $4 per foot to build, and the circular enclosure costs $ per foot to build. He will spend $300 on the two fences, but doesn t want the pens to take up a lot of space. What is the smallest total area he can enclose? (note: if it minimizes the area, it is an option for him to just build an enclosure for the cats and let the dolphins roam free. Or vice versa.) What is the largest possible area he can enclose? Calculating Antiderivatives Now that you re experts on finding derivatives, it s time to shift things around and go backwards find antiderivatives! Also, you should know about their brother, definite integrals. Besides the examples here, there are many, many examples of similar problems in your book and online, so if

4 you need more work, look around and find more problems. (3) Calculate the following indefinite integrals. (a) e x + 3x x dx (b) 2 πx dx (c) t(t + )dt (d) x+ x 2 +2x+ dx (e) ln x x dx (4) Calculate the following definite integrals. (a) 0 ex dx (b) xdx (c) x+ dx () What value of p would make the following equation true? 7 4 x 2 dx x 2 dx = p 7 x 2 dx (6) If an airplane s velocity is given by v(t) = 300t 2 miles per hour and it leaves Boston at 6:30pm, how far has it gone after hour? (7) Say a ball is thrown up into the air at an initial velocity of 0 m/s. Its acceleration is given by a(t) = 9.8m/s 2. How fast is it going after 3 seconds? How long before it hits the ground? (You can assume it was thrown from ground level.) (8) State the Fundamental Theorem of Calculus. Make sure you understand both parts. (9) Let s say you re given the following graph of f(x). What is 4 0 f(x)dx? (20) Let s say the derivative f (x) is given by the graph below. Suppose that f(0) = 3. What is f(2)?

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