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. You will be given exactly 90 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. You must indicate if you desire work on the back of a page to be graded. 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. I have read and understand the above instructions and statements regarding academic honesty:. SIGNATURE Page of
Standard Response Questions. Show all work to receive credit. Please BOX your final answer.. (a) (6 points) Find the most general antiderivative of f(x) = 4 cos x + 8. 4 sin x + 8x + C (b) (6 points) Evaluate: 5 7x 6 dx x 4 5 7x 6 x 4 dx = 5x 4 7x dx = [ 5 3 x 3 7x3] 3 = [ 5 ] [ 56 4 3 5 = 5 4 44 3 3 7 3 ] (c) (6 points) Let F (x) = x 3 t + dt. Find F (x). x 3 F (x) = ( F (x) = = 3x x 6 + t + dt (x 3 ) + ) (3x ) Page of
. (a) (8 points) A particle is moving along a line with acceleration (in m/s ) given by a(t) = 4t 3 + sin t. Given that the initial velocity is v(0) = 5 m/s, find the velocity at time t = π seconds. v(t) = t 4 cos t + C 5 = () + C (plug in 0 for t and 5 for v(0)) 7 = C v(t) = t 4 cos t + 7 v(π) = π 4 ( ) + 7 = π 4 + 9 m/s (b) (0 points) Use a linearization to find a good approximation of 9.0. Consider f(x) = x and a = 9. Then f(9) = 3 and So therefore giving us that f (x) = x f (9) = 9 = 6 L(x) = 3 + (x 9) 6 L(9.0) = 3 + (9.0 9) 6 L(9.0) = 3 + 600 9.0 Page 3 of
3. (a) (0 points) There are many curves y = f(x) which satisfy the following conditions: f is continuous, and the curve y = f(x) has a slant (or oblique) asymptote given by y = x f (x) > 0 for x (, ) (3, ), and f (x) < 0 for x (, 3) f (x) > 0 for x (, 3) (, ), and f (x) < 0 for x ( 3, ) Sketch the graph of one such curve below, making sure that all of the above conditions above are demonstrated by the curve you draw. Identify with a large dot the locations of any local max/min or points of inflection on your graph, and give the x-vaues of these points in the boxes below. 5 y 4 3 7 6 5 4 3 3 4 5 6 7 3 x Fill in this information: f has local minimums at these x-values: x = 3 f has local maximums at these x-values: x = 4 5 f has inflection points at these x-values: x = 3, x = (b) (8 points) Find the critical numbers (i.e., critical points) of the function f(x) = x 3/ + 6 x. f (x) = 3 x/ 3x 3/ = 3x/ = 3x 6 x 3/ 3 x 3/ So there are critical numbers when 3x 6 = 0 which is at x =. Note that x = 0 and x = are not critical numbers because they are not in the domain of f. Page 4 of
4. Suppose f(x) = x x +, f (x) = x (x + ), f (x) = (x3 3x) (x + ) 3. Answer the following questions or enter none in the case of no answer. (a) (4 points) Does f have symmetry about the y axis (even function), symmetry about the origin (odd function), both, or neither? Justify your answer. Yes f is odd since f( x) = ( x) ( x) + = x ( ) x x + = = f(x) x + (b) (7 points) Find the largest interval(s) where f is increasing and the largest interval(s) where f is decreasing. Express your answers using interval notation. f (x) = 0 when x = ± and f (x) is never undefined. Choosing test values we see: + + + f Therefore f is decreasing on (, ] [, ). Also f is increasing on [, ]. (c) (7 points) Find the interval(s) where f is concave up and the interval(s) where f is concave down. Express your answers using interval notation. f (x) = 0 when x = 0 and x = ± 3 and f (x) is never undefined. Choosing test values we see: + + + + + + f 3 0 3 Therefore f is concave down on (, 3) (0, 3). And f is concave up on ( 3, 0) ( 3, ). Page 5 of
5. (a) (8 points) Consider the problem of finding the point (x, y) that lies on the curve y = x + in the first quadrant and which is closest to the point (0, 3). (0, 3) (x, y) y = x + Define a function f of x which, if minimized, will give the x-coordinate of the point on the curve closest to (0, 3). Also give the domain of this function which is appropriate for the minimization problem. d(x, y) = (x 0) + (y 3) d(x) = (x 0) + ((x + ) 3) d(x) = x + (x ) f(x) = [d(x)] = x + (x ) (one possible answer) (another possible answer) Either way the domain is x > 0 (or x 0). (b) (0 points) Find the absolute maximum and absolute minimum values of on the interval [, ]. f(x) = x (x 8) f(x) = x 3 8x therefore f (x) = 6x 6x = x(3x 8) so f has critical points at x = 0 and x = 8/3 (not in the interval). So f( ) = 0 f(0) = 0 absolute max f() = 4( 4) = 6 absolute min Page 6 of
Multiple Choice. Circle the best answer. No work needed. No partial credit available. No credit will be given for choices not clearly marked. 6. (7 points) If the Mean Value Theorem is applied to the function f(x) = x x on the interval [, 4], which of the following values of c satisfies the conclusion of the theorem in this case? A. c = B. c = 3 C. c = D. c = 5 E. c = 3 7. (7 points) Using three equally-spaced rectangles of equal width, find the upper sum approximation of the area between the curve y = x and the x-axis from x = to x = 4. A. 8 B. 6 C. 4 D. 40 E. 48 8. (7 points) Evaluate A. B. 3π 4 3π 3 0 9 x dx. (Hint: a definite integral represents an area.) C. 9π 4 D. 9π E. 9π Page 7 of
9. (7 points) Which of the following is the equation of a horizontal asymptote for the curve y = 9x 5 x? A. y = 9 B. y = 9 C. y = 0 D. y = 5 E. y = 5 0. (7 points) 4 0 3 x dx = A. 6 B. 5 C. 4 D. 3 E.. (7 points) The graph of the first derivative f (x) of a function f(x) is shown. At what value of x does f have a local maximum? A. x = a B. x = b C. x = c D. x = d E. x = e a b c d e f (x) x Page 8 of
. (7 points) Which of the following definite integrals is equivalent to the following limit of a Riemann sum? n lim 8 + 5i n n 5 n i= 3 A. 8 + 5x dx 8 5 B. C. D. 8 + x dx 0 8 + 5x dx 0 5 0 5 8 + x dx E. None of the above. 3. (7 points) A farmer wants to build a rectangular pen which will be bounded on one side by a river and on the other three sides by a wire fence. If the farmer has 60 meters of wire to use, what is the largest area that the farmer can enclose? A. 00 m RIVER B. 400 m C. 450 m D. 600 m y x y E. 900 m 4. (7 points) Suppose A. 5 f(x) dx = 3 and 3 f(x) dx = 4. Find 5 3 f(x) dx. B. 7 C. D. 4 E. 8 Page 9 of
Congratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID ready so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score 8 3 8 4 8 5 8 6 8 7 8 9 Total: 53 No more than 50 points may be earned on the exam. Page 0 of
Algebraic FORMULA SHEET Limits a b = (a b)(a + b) 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 Trigonometric lim x a f(x) = L if for every ε > 0 there exists δ > 0 so that f(x) L < ε when x a < δ. lim x a f(x) exists if and only if sin θ lim = θ 0 θ cos θ lim = 0 θ 0 θ lim x a f(x) = lim x a + f(x) 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) (sin x) = cos x (cos x) = sin x (tan x) = sec x (sec x) = sec x tan x Theorems (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