Spring 2015 Sample Final Exam

Transcription

1 Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than what your actual final exam will be. Also I suspect that your final exam will have more short answer questions that this one. 1. Compute the following limits. Do NOT use L hôpital s Rule. (a) lim r 4 r 3 4r 2 r 2 2r 8 (b) lim x 4x x 3 (c) lim x 9 x 3 x 9

2 Math 1151 Sample Final Exam - Page 2 of 17 Final Exam on 4/30/14 2. Compute the following limits. You may use L hôpital s Rule. (a) sin x x lim x 0 x 3 (b) lim sin(x) ln(x) x 0 + (c) ( lim x ) 3/x x

3 Math 1151 Sample Final Exam - Page 3 of 17 Final Exam on 4/30/14 3. Use the limit definition of the derivative to compute the derivative of f(x) = 3 + 2/x. (x 0)

4 Math 1151 Sample Final Exam - Page 4 of 17 Final Exam on 4/30/14 4. A curve in the plane is given by the following equation x 2 + 2xy + 3y 2 = 4 (a) Find an expression for dy dx using implicit differentiation. (b) Find the equation of the tangent line to the curve at the point (2, 0). (c) Find the two points on the curve where the tangent line is horizontal.

5 Math 1151 Sample Final Exam - Page 5 of 17 Final Exam on 4/30/14 5. Find the derivative of f(x) = (ln x) 7x, x > 1 Find an equation of the tangent line to f(x) at x = e.

6 Math 1151 Sample Final Exam - Page 6 of 17 Final Exam on 4/30/14 6. Let f(x) = 3x 2/3 x. (a) Find all critical points. (b) For each critical point, determine if it is a local max, a local min, or neither. (c) Does the Mean Value Theorem apply to the function f on the interval [ 1, 27]? (d) Find all points c in ( 1, 27) where the instantaneous rate of change of f at c is equal to the average rate of change of f on [ 1, 27].

7 Math 1151 Sample Final Exam - Page 7 of 17 Final Exam on 4/30/14 7. Water is being pumped into an inverted conical tank at a constant rate of 1000π cubic centimeters per minute. The tank has a height of 300cm and a diameter of 200cm. [Note that the volume of a cone with height H and radius R is 1 3 πr2 H] How fast is the water level rising when the height of the water in the tank is 50cm?

8 Math 1151 Sample Final Exam - Page 8 of 17 Final Exam on 4/30/14 8. Graph a function with the following properties: 1. f is continuous on (, 4) ( 4, ). 2. f( 6) = 20, f(0) = 20, f(6) = 0 3. lim x 4 f(x) = 4. lim x 4 + f(x) = 5. lim x f(x) = lim x f(x) = f (x) < 0 for x in (0, ) 8. f (x) > 0 for x in (, 4) ( 4, 0) 9 f (x) < 0 for x in ( 4, 6) 10. f (x) > 0 for x in (, 4) (6, ) y x

9 Math 1151 Sample Final Exam - Page 9 of 17 Final Exam on 4/30/14 9. The graph below represents the velocity v(t) at time t of an object moving in a straight line. in m/s in s 2 (a) Use this graph to complete the following table specifying the location s(t) of the object at time t. t s(t) 0 (b) Determine the displacement of the object during the time interval [0, 8]. (c) Determine the distance that the object travels during the time interval [0, 8].

10 Math 1151 Sample Final Exam - Page 10 of 17 Final Exam on 4/30/ Consider the following sum n n n + 1 n n 2 n n n (a) Find a definite integral b a f(x) dx for which the above sum is a right Riemann sum. 2 n (b) Using the answer from part (a), find the limit of the above sum as n.

11 Math 1151 Sample Final Exam - Page 11 of 17 Final Exam on 4/30/ Evaluate the following integrals. (a) 4x x 2 dx (b) (8 + ln x) 3 dx x (c) π 0 d sin(t/2) dt 1 + t 2 dt

12 Math 1151 Sample Final Exam - Page 12 of 17 Final Exam on 4/30/ Let (a) Find G (2) G(x) = 1 4 u 3 x 3 x u 3 du (b) Find the equation of the tangent line to the graph of G(x) at x = 2.

13 Math 1151 Sample Final Exam - Page 13 of 17 Final Exam on 4/30/ Suppose that f(x) is continuous and differentiable everywhere. Below is a table of values for f(x) and f (x). Compute the following values. (a) Let g(x) = (x + f(x)) sin x. Find g (π). x f(x) f (x) 0 π 3 1 π/2 2 2 π/4 10 π 1-3 π/ π/4 6 5 (b) Let h(x) = f(tan 1 x). Find h (1). (c) Let H(x) = tan 1 (f(x)). Find H (π). (d) 1 0 sin(f(x)) f (x) dx (e) π/2 0 f (cos(x)) sin(x) dx

14 Math 1151 Sample Final Exam - Page 14 of 17 Final Exam on 4/30/ Consider the following piecewise function (c is some constant) (a) What is the domain of f(x)? f(x) = 8 x+5 for x 3 2 x for 5 x < 3 x 2 + c for x < 5 (b) Determine whether f(x) is continuous at x = 3. (c) For which value of c does the limit lim x 5 f(x) exist?

15 Math 1151 Sample Final Exam - Page 15 of 17 Final Exam on 4/30/ Use the linear approximation of f(x) = 3 x at a = 27 to approximate Let f(x) = 2x + e x. Note that f(x) is an one-to-one function and that f(1) = 2 + e. Find d f 1 (2 + e). dx

16 Math 1151 Sample Final Exam - Page 16 of 17 Final Exam on 4/30/ Let f(x) = x 1 + x 2. (a) Determine when f(x) is increasing and when it is decreasing. (b) Where is f(x) concave up and where is it concave down? (c) Find all local maxima and local minima of f(x). (d) Find all inflection points of f(x). (e) Find all horizontal asymptotes of f(x).

17 Math 1151 Sample Final Exam - Page 17 of 17 Final Exam on 4/30/ Find the largest possible area of a right triangle whose hypotenuse is 36 meters long.