2. Using the graph of f(x) below, to find the following limits. Write DNE if the limit does not exist:

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1 1. [10 pts.] State each of the following theorems. (a) [2 pts.] The Intermediate Value Theorem (b) [2 pts.] The Mean Value Theorem. (c) [2 pts.] The Mean Value Theorem for Integrals. (d) [4 pts.] Both parts to the Fundamental Theorem of Calculus. 2. Using the graph of f(x) below, to find the following limits. Write DNE if the limit does not exist: (a) (b) lim f(x) = x 1 + lim f(x) = x 1 (c) lim x0 + f(x) = (d) Find all x in [ 1, 3] at which f(x) is not continuous.

2 3. [6 pts.] For each of the following 2 functions on the specified intervals, if the Intermediate Value Theorem does NOT apply, state why. On the other hand, if it does apply, use it to prove that the function has a zero somewhere in the specified interval. (a) [3 pts.] f(x) = cos(2πx)2 x + cos(πx)3 x on [-1,1] (b) [3 pts.] g(x) = x2 + 1 x 2 x on [-1,1] (c) [3 pts.] g(x) = 5x 3 x + 1 on [-1,1] 4. [6 pts.] For each of the following 2 functions on the specified intervals, if the Mean Value Theorem does NOT apply, state why. On the other hand, if it does apply, find the point the theorem guarantees. (a) [3 pts.] f(x) = x 2 3 on [-1,1] (b) [3 pts.] g(x) = x2 4 x on [-1,1] 5. [7 pts.] 1. Find d x dx e ln(t) dt 2. Let F (x) = 1/x 0 cos(t 2 )dt. Find F (x).

3 6. [10 pts.] This problem must be done as stated. No credit will be given for using other methods. (a) [2 pts.] State the definition of the derivative of the function f(x) at the point x = a. (b) [5 pts.] Use part (a) to find the derivative of f(x) = x 2 + 3x + 1 ( or f(x) = 1/(2x + 2)) at the point x = 3. (c) [3 pts.] Use part (b) to find the tangent line to f(x) = x 2 + 3x + 1 (or f(x) = 1/(2x + 2)) at the point x = 3.

4 7. [12 pts.] For each of the following limits, compute it if it exists. If the limit does not exist, but tends to either ±, specify this, as well as stating that the limit Does not exist. (a) [2 pts.] lim x 1 x 2 x + 6 x 2 + x + 1 (b) [2 pts.] lim x 3 2e x 1 + e x (c) [2 pts.] lim x 3 x 2 + x + 1 x 2 6x + 9 (d) [2 pts.] ( lim x π ) sec(x) x π 2 2 (e) [2 pts.] lim x 0 1 cos(x) x 2 (f) [2 pts.] lim x 1 ln(x) x 0 (g) [2 pts.] lim x 0 9 x 15 x 2x (h) [2 pts.] lim x x 5 e x (k) [2 pts.] lim x x 2 x + 6 x 2 + x + 1

5 8. [30 pts.] Compute the following derivatives. (a) [6 pts.] d ( x 2 e x) dx (b) [6 pts.] f (t) for f(t) = sin(t) t (c) [6 pts.] dy 2 d 2 x for y = arctan(2x) (d) [6 pts.] g (x) for g(x) = π ln(x) + 2 x (e) [6 pts.] dr dt for r(t) = 3 sec(t2 t 1) [10 pts.] Find the tangent line to the graph of the equation x 2 y 3 + 2x = (x + y) at the point (1,2). 10. [10 pts.] Find the tangent line to the graph of the equation y = x x at the point x = 2.

6 11. [21 pts.] Compute the following. (a) [3 pts.] An antiderivative of the function f(x) = x 2 + x + 1 x + 1 x 2 (b) [3 pts.] The area under enclosed by the curves y = 4 x 2 and y = x + 2 on the interval [-2,3] (c) [3 pts.] The curve y which satisfies dy dx = 4x ex and passes through the point (0,2) (d) [3 pts.] The general form of the antiderivative of g(x) = 7 x (e) [3 pts.] t2 t t dt (f) [3 pts.] 2x sec 2 (x 2 + 2) dx

7 (g) [3 pts.] The average value of the function y = 3x 2 x 1 over the interval [1,3]. (h) [3 pts.] Compute π/4 0 2(1 + e tan(x) )sec 2 (x)dx. 12. [10 pts.] Compute the following. (a) [2 pts.] 5 i=3 60 i (b) [5 pts.] 100 k=10 2k 2 k + 1 (c) [3 pts.] Consider the sums S n = n i=1 1 n (i/n) 2 0. Expand (i.e. write out the numbers being added up) the sum for n = These sums are in fact right hand sums approximating a definite integral, that is lim n S n = 1 0 f(x)dx. What is this function f(x)? 2. What number do the sums S n approach as n gets large? 13. [10 pts.] Find all x-values at which the function f(x) given below is discontinuous: x 2, if 1 x < 3 f(x) = 13x 57 x 2 5x + 4, if 3 x < 5 x x 2 + 5, if 5 x < 7

8 14. [42 pts.] Do the following for the function f(x) = 2x 2 ln(x) 3x 2 (a) [6 pts.] Find lim x 0 f(x). (b) [6 pts.] Find lim x f(x). (c) [6 pts.] Find where f(x) is decreasing. (d) [6 pts.] Find all critical points of f(x). (e) [6 pts.] Find where f(x) is concave upwards. (f) [6 pts.] Find all inflection points of f(x). (g) [6 pts.] Sketch a graph of f(x) on the grid supplied on the next page. 15. [10 pts.] Find the global maximum and minimum of the function f(x) = 2x 3 + 3x 2 12x 6 on the interval [-1,2]. 16. [10 pts.] The Flying Dutchman is 8 nautical miles north of Ilsa Tortuga and heading there at a speed of 4 knots. At the same time, the Black Pearl is 6 nautical miles east of Ilsa Tortuga and heading away at 5 knots. How fast is the distance between them changing? 17. [15 pts.] During a mutiny, Captain Hendrick Vanderdecken is left with only a row boat 8 miles away from the closest point on the shore. If the nearest town is 4 miles down the shore from that point, what is the fastest he can get to town, provided he can row at a speed of 3 miles per hour and walk at a speed of 5 miles per hour?

9 18. [10 pts.] When he gets to town, Captain Hendrick Vanderdecken decides to make a poster to recruit a new crew. Since he s going to be nailing it up, he wants to give himself 1 inch margins on the top and bottom, and 2 inch margins on the sides. If he needs 50 square inches of printed material, what dimensions will need the least amount of paper?

AB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2

AB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2 AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6