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APPM 1350 Midterm 2 Spring 2012
On the front of your bluebook, please write: a grading key, your name, student ID, section, and instructor’s name (Chang or Guinn). This exam is worth 100 points and has 8 questions. Show all work! Answers with no justification will receive no points. Please begin each problem on a new page. No notes, calculators, or electronic devices are permitted.
- (12 points) Differentiate the following functions. Leave your answers unsimplified.
(a) y =
5 x 1 − x^3 (b) y = tan θ sec^2 θ
- (8 points) Use implicit differentiation to find dy/dx for the curve shown.
x
y
cosH x y L á y + 1
- (8 points)
(a) Find the linearization of f (x) = (1 + x)k^ at x = 0 for constant k.
(b) Use the linearization to approximate the value of 3
- (25 points) Consider the function f (x) =
x^2 4 − x^2 , f ′(x) =
8 x (4 − x^2 )^2 , f ′′(x) =
8(3x^2 + 4) (4 − x^2 )^3
(a) Find any vertical, horizontal, or slant asymptotes of f. Use appropriate limits to justify your answer.
(b) On what intervals is f increasing? decreasing?
(c) Find all local maximum and minimum values of f.
(d) On what intervals is f concave up? concave down?
(e) Find all inflection points of f.
(f) Using the information from (a)–(e), sketch a graph of f. Clearly label any points of interest, including any asymptotes, local extrema, and inflection points.
- (15 points) Water is draining at the rate of 48 m^3 /min from a shallow concrete conical reservoir of base radius 45 m and height 5 m. How fast is the water level falling when the water is 4 m deep?
- (8 points) A winter storm has left a coating of ice on a hemispherical dome. If the ice has a uniform thickness of 0. 2 cm and the dome has a diameter of 20 m, use differentials to estimate the amount of ice on the dome.
- (16 points) Farmer Joe has 20 , 000 m^2 (about 5 acres) of corn to plant and needs to build a new field. To manuever his tractor around the field, he needs to build a road around the perimeter of the field. The road needs to be 5 m wide along the length (x) of the field and 10 m wide along the ends (y) of the field so that he can turn the tractor around after driving down a row (see diagram).
Find the dimensions Farmer Joe should make his field (x and y) that minimize the amount of road he needs to build.
- (8 points) For each of the following, clearly write TRUE or FALSE (not just T or F). No justification is necessary.
(a) The function y = sin^2 x has horizontal tangents at x = 0, π/ 2 , and π.
(b) If h(x) =
2 f (x) + 3, f (2) = − 1 , and f ′(2) = 3, then h′(2) = 3.
(c) If f (5) = 6 and − 3 ≤ f ′(x) ≤ − 2 for all x, then the smallest possible value of f (9) is − 6 and the largest possible value of f (9) is 2.
(d) The function y =
3 x^2 − x + 1 x + 2 has a slant asymptote at y = 3x + 5.
