Math Fundamental Principles of Calculus Final - Fall 2015 December 14th, 2015

Transcription

1 Math Fundamental Principles of Calculus Final - Fall 2015 December 14th, 2015 Directions. Fill out your name, signature and student ID number on the lines below right now, before starting the exam! Also, check the box next to the class for which you are registered. You must show all your work and justify your methods to obtain full credit. Unless otherwise stated, please express your answers in calculator-ready form, meaning that you would be able to enter your final answer into a calculator. (You need not evaluate expressions such as ln 5, e 0.7, and 3.) Do not use scratch paper; use the back of the previous page if additional room is needed. Cell phones must be powered off and put away. No calculators are allowed, but you may use the sheet of notes that you brought with you. This may be no more than one sheet of paper. You may have anything written on it (on both sides), but it must be written in your own handwriting. Remember, USC considers cheating to be a serious offense; the minimum penalty is failure for the course. Cheating includes straying eyes and failing to stop writing when told to do so at the end of the exam. Good luck! Name (please print): Student ID: Signature: Check your section 9am MWF Appel 10am MWF Appel 9am MWF Blois 10am MWF Blois 1pm MWF Dreyer 2pm MW Dreyer 11am MWF Toulisse 12pm MWF Toulisse 12pm MWF Zhuang 1pm MWF Zhuang Do not write below this line! 1 6 (20pts) (20pts) 2 7 (15pts) (28pts) 3 8 (20pts) (15pts) 4 9 (18pts) (24pts) 5 10 (20pts) (20pts) 1st check 2nd check Total 1st check Total 2nd check

2 1. (20 pts) (a) Compute the following limits. If a limit is infinite, indicate whether it is + or. Write your answer in the box and show your work in the space below. (i) x 2 x 2 lim x 2 x 2 6x + 8 = (ii) lim x 1 x = x 1 (iii) 2 lim x e 1 x 1 = (b) Determine the value of the constant A so that the following function f(x) = Ax 2 x + 2 if x 1 4x + 4 x 2 2x 3 f(x) is continuous at x = 1. Justify your answer carefully! if x > 1 A = Page 2

3 2. (15 pts) A stationary store sells erasable pens. The store s profit P is a function of the selling price x of each pen, P = f(x). (a) Suppose that f (x) > 0. Which of the following statements must be true? (You do not need to explain your answer.) (A) The store will be making more profit tomorrow than it is today. (B) If they increase the price, their profit will increase. (C) If the store moves east, they will make more profit. (D) The graph of the profit function f(x) is concave-up. (b) Suppose that f (x) = x. The store makes a profit of $40 when the price x is $1. Use a linear approximation to estimate the new profit after the price is increased by 5 cents (to x = $1.05). (c) If f (x) = x, find the price at which the profit is maximized. Page 3

4 3. (20 pts) Find the equation of the tangent line to the curve at the point (x, y) = (1, 0). e xy = xy 2 + x Page 4

5 4. (18 pts) Shown below are the graphs of six functions. (A) (B) (C) (D) (E) (F) For each function above, choose which one of the graphs below is the graph of its derivative. Each graph (below) can be used once, more than once, or not at all. (1) (2) (3) (4) (5) (6) Fill in your answers in the table. Function A B C D E F Derivative Page 5

6 5. (20 pts) Consider the function with derivatives f (x) = (a) Find the domain of the function f(x). f(x) = x 1 (x + 1) 2. 3 x (x + 1) 3 and f 2(x 5) (x) = (x + 1) 4 (b) For each number a that is not in the domain of f, compute lim f(x) and lim f(x). x a + x a (c) Compute lim f(x) and lim f(x). x + x Page 6

7 (d) Find the intervals on which the function f(x) is increasing and on which it is decreasing. (e) Find the intervals on which the function f(x) is concave-upward and on which it is concave-downward, as well as the x-coordinates of any inflection points. Page 7

8 6. (20 pts) A closed box with a square base is constructed using steel on the bottom and the top, and wood on the sides. The steel costs $2 per square meter and the wood costs $4 per square meter. Find the dimensions of the box with the largest volume that can be constructed at a total cost of $16. Dimensions: Volume: Page 8

9 7. (28 pts) Calculate the following integrals: (a) 1 0 2x x3 + 3x + 1 dx = (b) e 2 1 x ln(x)dx = Page 9

10 (c) 1 0 (3x 1)e 3x dx = (d) ln(2) 0 e x e x e x dx = + e x Page 10

11 8. (15 pts) Consider the functions f(x) = (x + 1) 2 g(x) = 2x + 5 (a) Find the x-coordinates of all points where the graphs of these two functions, f and g, intersect. (b) Calculate the area of the region bounded by the graphs of f and g. Page 11

12 9. (24 pts) Consider the following function (a) Find all critical points of f. f(x, y) = x 3 + 9x x + y 3 3y (b) Classify each critical point that you found in part (a) as a relative maximum, a relative minimum, or a saddle point. Point Max/Min/Saddle Page 12

13 10. (20 pts) Compute the double integral 3x 5 e x3y da R where R is the rectangle given by 0 x 2 and 0 y 1. R 3x 5 e x3y da = Page 13