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MATH 131:100 Exam #3 Instructor: Keaton Hamm June 30, 2014 Print Last Name: First Name: Signature: “An Aggie does not lie, cheat or steal or tolerate those who do.” Instructions: You MUST clear all memory on your calculator: to do this hit 2nd, then + (MEM), then Reset (7), move the cursor over to the right ALL, select All Memory (1), then Reset (2) Show your work for each work out problem clearly and legibly. Box your final answer in the work out problems. Point values are shown for each problem. You may use a TI-83, TI-84, or TI-Nspire nonCAS version for all problems. Other calculators are not permitted. If you need extra scratch paper, ask me and I will provide you with some. Your grade will be written on the final page of the exam. There should be 8 pages with problems numbered 1-17 on this exam. There are 100 possible points. Good Luck! Section 1: Definitions Give the mathematical definition for each of the following terms or concepts. (Each problem is worth 4 Points) 1) State the Fundamental Theorem of Calculus 2) A number c is a Critical Point of a funciton f if 3) State the Mean Value Theorem 4) We say that f (c) is an absolute maximum of f if Section 2: Graphical Concepts 5) (12 Z xPoints, 2 each) The following is the graph of the function y = f (t). Let f (t)dt. g(x) = 0 y 00000000000000 11111111111111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 4 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 3 0 1 f(t) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 t 0 1 1 2 3 4 5 6 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 −1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 −2 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11111111111111 00000000000000 0 1 0 1 1 01 0 1 0 1 01 0 1 0 1 01 0 1 01 a) Evaluate g(0) b) Evaluate g(2) c) Evaluate g(6) d) Where is g increasing? e) Where is g decreasing? f ) Where does g have a local maximum value? Section 3: Work Out Problems 6) (6 Points) If f 00 (x) = x2 + 3 cos x f (0) = 2 and f 0 (0) = 3. Find the formula for f (x). 7) (5 Points) Suppose that f is a continuous function on [1, 10], and that f (1) = 5, f (10) = 7, f 0 (1) = 2, and f 0 (10) = 8. Evaluate Z 10 f 0 (x)dx 1 20 Z Z 30 f (x)dx = 5, 8) (5 Points) Suppose that 10 Then evaluate Z Z f (x)dx = 6, and 20 30 [3f (x) − 4g(x)] dx 10 9) (6 Points) Evaluate Z x2 + 3x − 1 dx x2 10) (6 Points) Evaluate Z 30 √ 1 3 + 2x − x dx x g(x)dx = 2. 10 11) (6 Points) If Z x3 g(x) = 2x2 t2 − 1 dt t+7 0 Find g (x). 12) (5 Points) Evaluate Z (ln x)2 dx x 13) (3 Points) Evaluate Z 2 −2 x + x3 − 5x7 dx x2 + 2 14) (6 Points) Find the absolute maximum and minimum of the function f (x) = −x2 + 4x − 3 on the interval [0, 4]. 15) (6 Points) Evaluate Z x2 (x3 + 5)9 dx 16) (10 Points) A rectangular box with an open top has a square base, and is made from 48 ft2 of material. What dimensions will result in a box with the largest possible volume? 17) (8 Points) Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. Final Score: