MATH 205 Final: MATH 205 Louisville Practice Final 150417
MATH 205–04 Practice Final Exam Name:
This test is closed-book and closed-notes. No calculator is allowed for this test. For full credit show
all of your work (legibly!), unless otherwise specified. For the purposes of this exam, all answers must
be in terms of familiar functions. Algebraic and trigonometric simplification of answers is generally
unnecessary.
1. (15 points) Answer the following questions.
(a) (5 points) What is the general antiderivative of x2−1
x2+1 −ex+ csc x(csc x+ cot x)?
(b) (5 points) If f′′(x) = 6x,f′(2) = 3, and f(1) = 0, find a formula for f(x).
(c) (6 points) Simplify the expression d
dx ∫x3
x2ln(t2
−7)dt.
2. (20 points) You have 1200 square centimeters of material with which to make a box with a
square base and an open top. Find the dimensions which maximize the volume of the box.
1/15
2/20
3/15
4/18
5/16
6/16
7/12
8/15
9/15
10 /8
11 /(6)
Σ/150
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MATH 205–04 Practice Final Exam
3. (15 points) The lemniscate is a curve satisfying the equation 2(x2+y2)2= 25(x2
−y2).
(a) (10 points) Find a formula for dy
dx on this curve in terms of xand y.
(b) (5 points) Find the equation of the tangent line to the lemniscate at the point (3,1).
4. (18 points) Evaluate the following integrals:
(a) (6 points) ∫4
0x√16 −x2dx.
(b) (6 points) ∫x3
x4+1 dx.
(c) (6 points) ∫π/3
0cos θ+ 2 sec θtan θdθ.
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Document Summary
For full credit show all of your work (legibly! For the purposes of this exam, all answers must be in terms of familiar functions. 1 (b) (5 points) if f (x) = 6x, f (2) = 3, and f (1) = 0, nd a formula for f (x). (c) (6 points) simplify the expression d dx x3 x2 ln(t2. 7)dt: (20 points) you have 1200 square centimeters of material with which to make a box with a square base and an open top. Find the dimensions which maximize the volume of the box. Practice final exam: (15 points) the lemniscate is a curve satisfying the equation 2(x2 + y2)2 = 25(x2. 0 x 16 x2dx. (b) (6 points) x3 x4+1 dx. (c) (6 points) /3. 0 cos + 2 sec tan d . Practice final exam: (16 points) a ten-foot-long ladder is leaning against a wall, with the base of the ladder six feet from the wall.
