Math 171 Spring 2017 Final Exam. Problem Worth

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1 Math 171 Spring 2017 Final Exam Problem Worth Total Last Name: First Name: Student ID: Section: Remember to show all of your work and provide all necessary explanations for full credit. Good luck! Helpful formulas: nn nn kk = kk=1 kk 2 = kk=1 nn(nn + 1) 2 nn(nn + 1)(2nn + 1) 6

2 Question 1. (3 points each) Compute the following limits. If any limit equals or, you must say so explicitly. Does not exist will not suffice for full credit. Any uses of L Hôpital s rule must be justified for full credit. A. lim tt 1 ln (tt) B. lim xx 2 xx xx 2 C. lim xx2 9 xx 3 xx 3

3 Question 2: (3 points each) Compute the following limits. If any limit equals or, you must say so explicitly. Does not exist will not suffice for full credit. Any uses of L Hôpital s rule must be justified for full credit. A. lim xx xx2 +2xx+3 3xx 2 +2xx+1 sin (ππππ) B. lim xx 0 xx

4 Question 3. (3 points each) Let Find: ff(xx) = 5xx2 xx 4xx 2 2 (a) The horizontal asymptotes of ff. (b) The vertical asymptotes of ff.

5 Question 4. (5 points) Determine the value of the constant aa for which the function xx 2 + xx 2 ff(xx) =, if xx 1 xx 1 aa, if xx = 1 is continuous at 1.

6 Question 5. (3 points each) Compute the following derivatives A. dd dddd 2 sin(xx) eexx + xx 7 1 xx B. dd dddd (xx3 ln(2xx)) C. dd dddd xx 1+xx 2

7 Question 6. (4 points each) Compute the following derivatives A. dd dddd sin 1 (xx 2 ) B. dd dddd ee xx

8 Question 7. (5 points) The function ff(xx), defined on [-2,2], is depicted in the figure below. Sketch on top of this figure the function ff.

9 Question 8. Consider a function ff. The graph of the first derivative ff as a function xx is shown. (a) (4 points) According to ff, on what intervals is ff decreasing? (b) (2 points) At what values of xx does ff have a local maximum? (c) (2 points).. At what values of xx does ff have a local maximum?

10 Question 9. (8 points) Sketch a graph of yy = ff(xx) on the grid below that passes through the points ( 4,0), (0,0), and (4,0) indicated by black dots on the grid. Your graph should satisfy the following conditions for ff and ff : ff (xx) > 0 for xx > 3 and 3 < xx < 0 ff (xx) < 0 for xx < 3 and 0 < xx < 3 ff (xx) > 0 for xx < 2 and xx > 2 ff (xx) < 0 for 2 < xx < 2

11 Question 10. (8 points) The area of a circle increases at a rate of 1 cm 2 /s. How fast is the radius changing when the radius is 2 cm? Hint: the area AA of a circle is related to its radius rr by AA = ππrr 2.

12 Question 11. Consider gg(xx) = xx 2 16 on the domain (0, ). (a) (3 points) Find the critical points of gg. xx (b) (4 points) Find the intervals on which gg is increasing or decreasing. (c) (3 points) Use the first derivative test to determine whether each critical point is a local maximum, a local minimum or neither.

13 Question 12: Complete the following steps to calculate a Riemann sum for the function ff(xx) = xx + 8 on the interval [0, 6] with nn = 3. A: (2 pts) Sketch the graph of ff on the given interval (make sure to label the axes). B: (2 pts) Find xx and the left-hand endpoints xx 0, xx 1, xx 2. C: (2 pts) Illustrate the left Riemann sum (the sum using left-hand endpoints) on your sketch above. D. (2 pts) Calculate the Riemann sum.

14 Question 13. (5 points) Evaluate the following expression. Do not attempt to simplify. (Hint: You may wish to use the summation formulas on the front of the test.) 99 (kk + 1) 2 kk=1

15 Question 14. (5 points) The figure shows regions bounded by the graph of ff and the xx-axis on the interval [ 4,4]. Evaluate the following integral. 4 4 ff(xx)dddd

16 Question 15 (3 points each) Evaluate each indefinite integral. A. (xx xx + 1 2xx ) dddd BB. (cos(tt) sec 2 (tt)) dddd

17 Question 16 (3 points each) Evaluate each indefinite integral. A. (3xx + 1)(4 xx)dddd B. 3tt2 +4tt 5tt dddd

18 Question 17. At time tt = 0, a ball is popped up vertically (from the ground) with a velocity of 30 meters/second. Assume the acceleration due to gravity is aa(tt) = dddd dddd = 10 meters/second A. (3 points) Find an expression for the velocity vv of the ball, as a function of time, after the ball pops up, but before it hits the ground. B. (3 points) Find an expression for the height ss of the ball, as a function of time tt, after the ball pops up, but before it hits the ground. C. (2 points) At what time after the ball pops up does it hit the ground?

19 Question 18. (4 points each) Evaluate each definite integral. 0 A. 3ee xx dddd 1 B. xx 1 3 dddd 8 1

20 Question 19 (2 points each) Evaluate the following derivatives: A. dd xx dddd 5 tt2 + tt + 1 dddd B. dd 5 dddd tt2 + tt + 1 dddd xx C. dd xx 2 dddd 5 tt2 + tt + 1 dddd

21 Question 20 (3 points) Use symmetry or the properties of integrals to evaluate or help evaluate the following expressions A. ee xx2 dddd + ee xx2 dddd C. (1 sin 3 (xx)) dddd ππ ππ

22 Question 21 (3 points each) Use a change of variables (uu-substitution) to evaluate each of the following indefinite integrals. A. (xx 2 + 1) 10 2xx dddd B. xx ee xx2 dddd

23 Question 22. (6 points) Use a change of variables (uu-substitution) to evaluate each of the following definite integrals. ππ/2 A. sin 2 (θθ) cos(θθ) dddd 0 B. ππ/ xx 2 dddd

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