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For each part, clearly indicate your final short answer in the appropriate BOX provided. [12 marks, 2 mark for each part, with NO part marks for incorrect final answers.] Here is the graph of the function g(x)== 7. (NOTE: the graph is not drawn to scale.) * 18(x)=2 3+e? (NOTE - - - - - - - - - - - - - - - - Use the graph and/or the formula defining g to answer the following questions (no justification is required, you should show any work needed). a) Is x = 0 a vertical asymptote? (Circle the correct answer in the box provided.)

(b) We want to estimate a solution to the equation 24 = + 7 Find a positive integer n such that a solution lies in the interval [n, n + 1]. Answer:

b) Is there a function f(x) such that lim f(x) does not exist but f(2) = 7? If there is no such function, explain why. Otherwise, give an example by sketching a graph of what such a function would look like. (3 marks) Yes FIA 71 jump discontinuity at X2. y= f(x) 2 l of - 1 > X X^2+ Tim fix=7+2=lim, flx) X2 but f(2)=7 Ansters may

[10] 6. Let W = span{(0,1,0,1),(1,0.0.1).(1.-1.0.0)}. Let X = (1, 1.0.1). Find the or- thogonal projection of X onto W.

4. (20 pts) A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft /s, how fast is the angle between the top of the ladder and the wall changing when the angle Follow the indicated steps. rad? STEP 1: Make a sketch and label it. STEP 2 : Solve the problem.

1. 22 sin(4x) 2 1+0 sin(2x) sinº (3x) = cult Win 0 tonn

A company makes two products, A and B, whose joint demand functions are 9. = 90 -4p. -2p. 9,- 50-2p. --2p. b) Show that the total revenue function for this company is r = 90p. -4pi -4P.Pa +50p, -2pg?

8. (10 marks) (a) Write out the third-order Maclaurin polynomial for the function In(1+r). (b) How many terms in the Maclaurin polynomial should you keep if you wish to approximate In(1.1) with an error below 10-6?

1 [ 2a d d+gl 25. Find det 26 e eth 2c f fti mark

3. (18 pts) MULTIPLE CHOICE ! ! ! A table of values for f(x) and f'(x) is shown below. Suppose that f is a one-to-one function and f (x) is its inverse. f' x 1 3 4 f (x) 3 4 6 (x) 4 5 3 CIRCLE THE CORRECT ANSWER IN EACH PART. (I) Evaluate f-1 (f (x)) at x = 3. (a) 1 ; (b) 3; (c) 6; (a) 4; (e) DOES NOT EXIST; (£) NONE OF THE PREVIOUS ANSWERS. (11) Evaluate e f (f (x)) at x = 3. (a) 6 ; (b) 25; (c) 5; (d) 15; (e) DOES NOT EXIST ; () NONE OF THE PREVIOUS ANSWERS. (111) Evaluate e In (f (x)) at x = 3. (e) DOES NOT EXIST; (f) NONE OF THE PREVIOUS ANSWERS. (IV) Evaluate f-1 (x) at x = 3. (a) 4; (a) 4 ; (b) 1; (b) 1 ; (com () 5; (d) 5; (e) DOES NOT EXIST; () NONE OF THE PREVIOUS ANSWERS. (v) Evaluate -? (x) at x = 3. (a) 1 ; (b) 4; (e) 5; () NONE OF THE PREVIOUS ANSWERS. (VI) Find the average rate of chnage of f over the interval (1, 3). (a) 2 ; (b)1; (c) ; (a) 5; (e) DOES NOT EXIST; () NONE OF THE PREVIOUS ANSWERS.

The diagram at right shows a sernicircle of radius 1, subdivided into two sectors AOC and COB (b) For what value of 0 is the product of the two areas maximised? You must support your answer with clear reasoning and calculations.

2. Consider the 3 x 3 matrix A= ſo 1 0 1 1 -1 -1] 1 1 (a) Find the eigenvalues of A and find a basis of each of the corresponding eigenspaces. (b) Is A diagonalizable? Justify your answer.

Problem 8 a) Use your calculator and left Riemann sums for n = 4 subintervals of equal length, to approximate the area bounded by the curve f(x) = x and the z-axis on the interval [2, 4]. Follow the steps below. i) (10 points) Find A.r, the grid points Ik, and fill in the table. If(tk) Ax= ii) (6 points) Compute the left Riemann sum with two decimal places. b) (12 points] A rectangle is constructed with its base on the x-axis, and its two top vertices on the parabola y = 25 - 22. What are the dimensions of the rectangle with the maximum area?

9. Answer the following questions about the function g(1) = cos(2x + 7). (a) What is the amplitude of g(x).

(b) How many terms in the Maclaurin polynomial should you keep if you wish to approximate In(1.1) with an error below 10-6?

1. Let f(x) = V4+* (> 0). [3] (a) Sketch the graph of y = f(x). Use your graph to explain why f(x) is one-to-one. 3] (b) Find an expression for f-1(2) and clearly state its domain. [4] (c) Find an equation for the tangent line to the graph of y=f-1(x) at the point x = 3.

10. In all of this question let f(x) = -2? + 4.c. (a) Make a small sketch of the region bounded by the r-axis and the curve y = f(1) and then use the Fundamental Theorem of Calculus to find its area. [5 points) (b) Use the definition of the definite integral to find the area of the region in Part (a). Show all of your work. (11 points)

5. The length of a rectangle is decreasing at the rate of 1 foot per second, but the area remains constant. How fast is the rectangle's width is changing when its length is 10 feet and its width is 5 feet? (10 points) A=lw 4A = 10 + antw. Since A is a constant A = 0 0 = 10 + ( - 1)5 = lot = -

(b) If the current demand is q = 3, will the revenue increase or decrease if the price is raised slightly?

5. Which of the below limits are equal to on? etetve 2+2+ x2 – 2.r ) X X -2 tve te B. lim tan - 2 + /2+ V tye 22 - 25 C +5 lim lin ( 2+3+ 22 - 3 *337 Xlx-3) Dling – In(sin x) lim 22 - 25 r2- 3. - lih +3 lim ks)

4. Find all solutions to the following: (b) 3 sin2 – 6 sin’ x + 3 = 0

For each part, clearly indicate your final short answer in the appropriate BOX provided. [12 marks, 2 mark for each part, with NO part marks for incorrect final answers.] Here is the graph of the function g(x)== 7. (NOTE: the graph is not drawn to scale.) * 18(x)=2 3+e? (NOTE - - - - - - - - - - - - - - - - Use the graph and/or the formula defining g to answer the following questions (no justification is required, you should show any work needed). d) Does the function have an absolute maximum? (Circle the correct answer in the box provided.)

13. The area of the region enclosed by the curve y =lnx, y=0 and r4 is A) 41n 4-3 B) In 4 C) 41n 4-4 E) none of these

11. Let f(1) = cos 1. Use the definition of the derivative to prove that f'(:1) = -sin . Solution: We have · f'(:1) = lim cos(2+h) - cos(2) +0 cos(1) cos(h) - sin() sin(h) - cos(1) = lim h0 sin / – Jin contar) (cos(1) = -1) = sin(e) (sinm) h0 sin(h) cos(h) - = lim cos(2) h cos(h) - 1 = cos(x) lim h- oh = cos(x)0 - sin(x)(1) = - sin h c) lim 0 / h

11. A die is weighted so that each odd number, 1-3-5, has the same probability of occurrence. Each odd number is three times as likely to appear as each even number, 2-4-6. What is the probability of the event E= {2,4,6)? (A)} (B) (C) (D)} (E)

PROBLEM 13: Your wallet contains two $1 bills, one $5 bill, and one $10 bill. You need $4 to pay for lunch, so you begin selecting bills from the wallet, one after another and without replacement, and noting the value of each bill. You stop as soon as the total value of the bill(s) selected is at least $4. How many different outcomes are possible? A) 5 B) 6 C) 4 D) 7 E) 24 N) none of the above

23. A survey of IU students reveals that 10% of the fresh- men, 15% of the sophomores, 60% of the juniors, and 85% of the seniors like Finite Math. Given that 30% of all students are freshmen, 30% are sophomores, 20% are juniors, and 20% are seniors, what percent age of the student population like Finite Math? (a) .635 (b) .535 (c) .465 (d) .365 (e) 235 (f) none of the above