MLC Practice Final Exam

Transcription

1 Name: Section: Recitation/Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response questions. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. You will be given exactly 0 minutes for this exam. Remove and utilize the formula sheet provided to you at the end of this exam. ACADEMIC HONESTY Do not open the exam booklet until you are instructed to do so. Do not seek or obtain any kind of help from anyone to answer questions on this exam. If you have questions, consult only the proctor(s). Books, notes, calculators, phones, or any other electronic devices are not allowed on the exam. Students should store them in their backpacks. No scratch paper is permitted. If you need more room use the back of a page. Anyone who violates these instructions will have committed an act of academic dishonesty. Penalties for academic dishonesty can be very severe. All cases of academic dishonesty will be reported immediately to the Dean of Undergraduate Studies and added to the student s academic record. This is a practice exam. The actual exam may differ significantly from this practice exam because there are many varieties of problems that can test each concept. Score: /08 (no more than 00 points may be earned on the exam.) Page of

2 Standard Response Questions. Show all work to receive credit. Please BOX your final answer.. A ball is thrown upward in the air. Suppose the velocity of the ball, in feet per seconds, is given by v(t) = 3 3t, where t is in seconds. (a) (4 points) When does the ball reaches its maximum height? (b) (4 points) If the ball is thrown from an initial height of feet, find the precise height function h(t) of the ball. (c) ( points) What is the maximum height reached by the ball? (d) ( points) What is the acceleration of the ball after second? Page of

3 . (6 points) Using the limit definition of the derivative, find f (x) of f(x) = x (6 points) Show that the equation x x + x + 3 = 0 has at least one real solution. Hint: Quote and use the Intermediate Value Theorem. You have to show why it applies. Page 3 of

4 4. Consider the functions f(x) = x and g(x) = x. (a) ( points) Sketch the graphs of y = f(x) and y = g(x) below on the same set of axes AND shade in the region between (bounded by) these curves. (b) (7 points) Set up BUT DO NOT EVALUATE an integral that will find the area A of the region between the curves y = f(x) and y = g(x). Page 4 of

5 . Consider the function f(x) (as well as its first two derivatives) given below: f(x) = f (x) = x f (x) = 6(x ) 3 + x (3 + x ) (3 + x ) 3 (a) ( points) Find all vertical and horizontal asymptotes of f(x) (if there are any). Be specific about which type of asymptote(s) you have found and which types do not exist. (b) ( points) Find all x-intercepts and y-intercepts of f. (c) (3 points) Find all critical numbers of f (if there are any), find the x-intervals where f is increasing and decreasing, AND find the x-values where f has local maxima and minima (if there are any). (d) (3 points) Find the x-intervals where f is concave up and concave down AND find the x-values of the inflection points of f (if there are any). (e) ( points) Combining ALL previous parts, draw an accurate sketch of y = f(x) below. ALSO, label any asymptotes, intercepts, local maxima/minima, and inflection points. Page of

6 6. ( points) The area of an equilateral triangle is increasing at a rate of 3 square inches per second. At what rate will the height of the triangle increase when the length of the sides is 7? Page 6 of

7 7. ( points) A rectangular fence is to be built from BOTH wood AND metal so that opposite parallel sides of the fence are made from the same type of material. The wood costs $0 per foot of fence used and the metal costs $0 per foot of fence used. If you only have $400 to spend on fencing material, what dimensions of the fence will enclose the largest area? Page 7 of

8 Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearly marked. 8. (3 points) Evaluate the following limit: lim y 6 y 4 y 6 A. /4 B. /4 C. /8 D. DNE 9. (3 points) Evaluate the following limit: lim z 0 (z + 9) z 3z A. DNE B. 3 C. 3 D. /3 0. (3 points) Evaluate the following limit: lim t sin(t ) t A. B. C. D. Page 8 of

9 . (3 points) Find the following derivative: A. x 3 ( x 3 + cos(x)) B. x + ( 3 )( x 3 + cos(x)) C. x 3 ( x + sin(x)) ( x 3 + cos(x)) [ ( ) ] 3 d x x + sin(x) dx D. x( x + sin(x)) 3 + x ( x + sin(x)) ( x 3 + cos(x)). (3 points) Find the following derivative: A. sin ( ( t+ B. sin C. sin ( D. sin ( ( t+ ( ) ( t+ ) ( ) ) ) ( ) ) ) ( t+ ) ) [ ( )] d t + cos dt 3. (3 points) Find the following derivative: A. sin(x + )(x) B. cos(x + )(x) C. sin(x + ) D. cos(x + ) [ d x ] sin(t + ) dt dx Page 9 of

10 4. (3 points) Evaluate the following integral: A. 4 B C. 3 6 D. 7 0 (x + x) dx. (3 points) Evaluate the following integral: A. 0 t /3 + 3 t dt B. ( 7 4 )3/ C. ( 7 4 )3/ D (3 points) Evaluate the following integral: A. B. 3 C. 3 6 ( ) π 6 π D. 6 π /4 π /6 cos( θ) sin ( θ) θ dθ Page 0 of

11 7. (3 points) Find the slope of line tangent to the curve C : xy 3y + x 3 = 4 at (, ). A. 8 B. C. D. 0 x + 8 if x < 8. (3 points) Consider the function f(x) = 3 if x = x + a if x > x If possible, find the value of a for which the function f(x) above is continuous at x =. A. 0 B. C. D (3 points) Estimate using linearization. A. 3 8 B C. 7 4 D. 9 4 Page of

12 FORMULA SHEET a b = (a b)(a + b) Algebraic a 3 b 3 = (a b)(a + ab + b ) Quadratic Formula: Area of Circle: πr b ± b 4ac a Geometric Circumference of Circle: πr Circle with center (h, k) and radius r: (x h) + (y k) = r Distance from (x, y ) to (x, y ): (x x ) + (y y ) Area of Triangle: bh opposite leg sin θ = hypotenuse adjacent leg cos θ = hypotenuse opposite leg tan θ = adjacent leg If ABC is similar to DEF then AB DE = BC EF = AC DF Volume of Sphere: 4 3 πr3 Surface Area of Sphere: 4πr Volume of Cylinder/Prism: (height)(area of base) Volume of Cone/Pyramid: 3 (height)(area of base) sin θ + cos θ = sin(θ) = sin θ cos θ cos(θ) = cos θ sin θ = sin θ = cos θ Table of Trig Values Trigonometric x 0 π/6 π/4 π/3 π/ sin(x) 0 / / 3/ cos(x) 3/ / / 0 tan(x) 0 3/3 3 DNE Limits lim f(x) exists if and only if lim f(x) = lim f(x) x a x a x a + sin θ lim = θ 0 θ cos θ lim = 0 θ 0 θ Derivatives f f(x + h) f(x) (x) = lim h 0 h (fg) = f g + fg ( ) f = f g fg g g (f(g(x))) = f (g(x)) g (x) Trig derivatives (sin x) = cos x, (tan x) = sec x, (sec x) = sec x tan x, Theorems (cos x) = sin x (cot x) = csc x (csc x) = csc x cot x (IVT) If f is continuous on [a, b], f(a) f(b), and N is between f(a) and f(b) then there exists c (a, b) that satisfies f(c) = N. (MVT) If f is continuous on [a, b] and differentiable on (a, b) then there exists c (a, b) that satisfies f f(b) f(a) (c) =. b a (FToC P) If F (x) = then F (x) = f(x). x a f(t) dt Other Formulas Newton s Method: x n+ = x n f(x n) f (x n ) Linearization of f at a: L(x) = f(a) + f (a)(x a) n c = cn i= n i = i= n i = i= n(n + ) n(n + )(n + ) 6 Page of