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Mat 270 Final Exam Review (2.1-5.5) Sheet Spring 2012 (Final on April 26, 7:10 - 9:00 PM in PSY 102)
1. Find the slope of the secant line to the graph of [pic]between the points [pic]and [pic]. Answer: [pic]
2. Find the equation of the secant line to the graph of [pic]between the points at [pic]and [pic]. Answer: [pic]
3. Find the equation of the tangent line to the graph of [pic]at the points [pic].
Answer: [pic]
4. Find the equation of the normal line to the graph of [pic]at the points [pic].
Answer: [pic]
5. Use conjecture method to evaluate the limit [pic] at x = 9. Answer: 6
6. Evaluate the following limits using algebraic process:
a) [pic] Answer: 6
b) [pic] Answer: [pic]
c) [pic] Answer: [pic]
d) [pic] Answer: [pic]
e) [pic] Answer: [pic]
f) [pic] Answer: [pic]
g) [pic] Answer: [pic]
h) [pic] Answer: [pic]
i) [pic] Answer: [pic]
j) [pic] Answer: [pic]
7. Find vertical and horizontal asymptote(s) if any: [pic]
Answer: VA: [pic] and HA: [pic]
8. Find vertical and horizontal asymptote(s) if any: [pic]
Answer: VA: [pic] and horizontal asymptote HA is [pic].
9. Write the limit definition of derivative for the function [pic] at x = a
10. Use limit definition of derivative to find [pic] of the function [pic]
11. Find the equation of the tangent line to [pic] at the point (1, 1)
12. Find the equation of the tangent line to [pic] at the point [pic]
Answer: [pic]
13. Where is the slope of the tangent line to the curve [pic] equal to –24?
At [pic]
14. Find the derivative [pic]:
1) [pic] Answer: [pic]
2) [pic]
3) [pic]
Answer: [pic]
4) [pic]
5) [pic] Answer: [pic]
6) [pic] Answer: [pic]
7) [pic]
8) [pic] Answer: [pic]
9) [pic] Answer: [pic]
10) [pic]
11) [pic]
12) [pic]
13) [pic]
15. Find the equation of the tangent line to [pic]at x = 3 where [pic] Answer: [pic]
16. Find the equation of the tangent line to [pic] at x = 2
17. Find the equation of the tangent line to [pic] at [pic]
18. Find the equation of the normal line to [pic] at [pic]
19. The slope of [pic] at [pic] is [pic], find [pic]. Answer: [pic]
20. The function [pic] is one-to-one for [pic]. Find [pic]
21. Find [pic], the derivative of the inverse function of f if [pic].
22. A cost function of the form [pic] reflects diminishing returns to sale. Find and graph the cost, average cost and marginal cost functions.
23. An 8 foot ladder is leaning against a wall. How fast is the top of the ladder sliding down the wall if the bottom of the ladder is sliding directly away from the wall at 2 ft/sec and the foot of the ladder is 2 ft away from the wall? Answer: [pic]
24. A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. Given the surface area [pic]. Hint: Volume of a sphere is [pic] and [pic] where k is a constant. Now find that [pic], shows that rate of change of radius is a constant.
25. Baseball runners stand at first and second base in a baseball game. At the moment a ball is hit, the runner at first base runs to second base at 18ft/sec; simultaneously on the second base runs to the third base at 20ft/sec. How fast is the distance between the runners changing 1 second after the ball is hit? The distance between consecutive bases is 90 ft and the bases lie at the corner of a square.
26. Find all relative/absolute max/min, critical point(s), interval of increase/decrease, inflection point(s) interval of concavity:
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
27. Determine the open interval where the function [pic] is concave up.
28. An 8 ft tall fence runs parallel to the side of a house 3 ft away. What is the length of the shortest ladder that clears the fence and reaches the house? Assume that the vertical wall of the house and the horizontal ground have infinite extend. Answer: 15 ft. (see example 4 at page number 260)
29. Find the dimensions of the right circular cylinder of maximum volume that can be placed inside the sphere of radius R.
30. A 5324 cubic foot tank with a square base and an open top is to be constructed of a sheet of steel of a given thickness. Find the length of a side of the square base of the tank with minimum surface area. Answer: Length of side of square base is 22 ft.
31. Linear Approximation:
1) Approximate [pic] using the linear approximation of [pic] at x = 0
Answer: [pic]
2) Approximate [pic] using the linear approximation of [pic] at x = 1
Answer: [pic]
32. Verify the conditions of Mean Value Theorem or Rolle’s Theorem and find the value of c in the open interval given.
1) [pic]
2) [pic]
33. Evaluate the following integrals:
1) [pic]
2) [pic]
3) a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
11) [pic]
12) [pic]
13) [pic]
14) [pic]
15) [pic]
16) [pic]
17) [pic]
18) [pic]
19) [pic] Answer: [pic]
20) [pic] Answer: [pic]
21) [pic] Answer: [pic]
22) [pic] Answer: [pic]
23) [pic] Answer: [pic]
24) [pic]
25) [pic]
26) [pic]
27) [pic]
28) [pic]
34. Evaluate the following:
1) [pic] Answer: [pic]
2) [pic] Answer: [pic]
3) [pic] Answer: [pic]
4) [pic] Answer: [pic]
35. Given [pic], find [pic]
36. Find LHS, RHS and Midpoint sum (MPS) use Riemann sum:
1) [pic]
2) [pic]
3) [pic]
4) [pic]
37. Evaluate the sum: [pic]
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