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Math 160. Test 4. (Harvey Spring 2005)
Name: (2 points)
No notes or texts allowed. You may use a TI-83, TI-84, TI-86 or equivalent calculator. Show all work.
1-3 (7 points each) Compute the indefinite integrals.
- (^) ∫ (x^3 + 2x + 1) dx
5 ex^ + x +
dx
(3x + 1)^2 dx
4-6 (7 points each) Compute the derivative f ′(x).
f (x) = x ln(x)
f (x) = e(x^3 +x)
f (x) = e
x x + 1
7 (6 points). Solve for x: log 3 (x + 4) = 2
8-11 (6 points each) In problems 8-11 we consider the function:
f (x) = e−x^2 − 1
- Identify all intercepts and asymptotes of f (x).
9 Calculate and simplify f ′(x). Identify all the critical points of f (x).
10 Calculate and simplify f ′′(x). Identify all the inflection points of f (x).
11 Sketch the graph of f (x) labeling all relevant data gathered in the previous steps.
12 (10 points) Archeologists recently excavated finely chipped spear points from a nomadic hunt- ing site in Lubbock, Texas. Bison bone fragments at that site were analyzed. In a sample that would initially have contained 3.6 grams of C^14 , only 1.1 grams remained. Based on this data, how old are the spear points? (Recall that the half life of C^14 is 5730 years).
13-14 (10 points each) Work two of the remaining three problems. Indicate which two you would like me to grade by placing check marks in the appropriate boxes.
13 A rectangle is inscribed in a right triangle, as shown in the figure. If the triangle has sides of length 5, 12, and 13, what are the dimensions of the inscribed rectangle of greatest area?
12
5
13
14 A farmer wants to build a fence to enclose a rectangular area of 8,000 square feet. In addition, there will be two internal dividers. If it cost $4 dollars per foot for the exterior fence, and $3 dollar per foot for the interior partitions, what dimensions will minimize the costs?
15 An open top box is formed by cutting the corners from a 24 × 24 square and folding up the sides. What size corners should be removed in order to maximize the volume of the box?
solutions
=
x^4 4 +^ x
(^2) + x + C
A = x · y =
5 y
y = 12y −
5 y
2 =⇒ A′ = 12 − 24
5 y
A′^ = 0 : 24 5
y = 12 =⇒ y = 5/2 =⇒ x = 6
xy = 8000 =⇒ y = 8000/x
C = 14y + 8x = 14
x
- 8x =⇒ C′^ = − (^112000) x 2 + 8
C′^ = 0 : 8 x^2 = 112000 =⇒ x = 118.32 =⇒ y = 67. 61
- Let x be the length of a square cut from the corner.
V = (24 − 2 x)^2 x = 576x − 96 x^2 + 4x^3
V ′^ = 576 − 192 x + 12x^2 V ′^ = 0 : 48 − 16 x + x^2 = 0 =⇒ (x − 12)(x − 4) = 0
Cut 4 × 4 corners from the square.
