Description
1. (18 points)
Compute the following limits, if they exist. Else, argue why the limit does not exist.
4 (a) limx→−2
1
x −
1
2
x
3 − 8
7 (b) limy→∞
e
−y
sin(y) cos(y)
y
7 (c) limr→1
|r−1|
2r−2
Exam, Page 1 of 5
Name: Calculus and Linear Algebra I
2. (16 points)
6 (a) Show that the equation x
6 − 5x − 5 = 0 has at least one solution on the interval
[−1, 0].
10 (b) Compute the derivative of f(x) = 1
x2 directly from its definition as the limit of a
difference quotient.
Exam, Page 2 of 5
Name: Calculus and Linear Algebra I
3. (20 points)
Consider the function f(x) = x
2
2 − x
2
.
What is the domain of f? Find the horizontal and vertical asymptotes, local minima, local maxima, and reflection points of f. Identify the regions where the graph of f
is concave up or concave down. Finally, sketch the graph.
Exam, Page 3 of 5
Name: Calculus and Linear Algebra I
4. (32 points)
Solve the following:
10 (a) Integrate R
x + 1
x
2
(x − 1)dx
6 (b) Integrate R
x
−3
e
−1/x2
dx
8 (c) Integrate Z 2π
0
e
x
cos(x) dx
8 (d) Differentiate f(t) = t
t
3
Exam, Page 4 of 5
Name: Calculus and Linear Algebra I
5. (14 points)
Choose only two of the below:
7 (a) Find the area between the curves x = y
2 and 0 = −x − y
2 + 2 (in absolute terms).
7 (b) A farmer owns an 8 km long stretch of land between two parallel rivers that are
1500 m apart. What is the area of the largest rectangular enclosure he can fence off
with (i) 1 km of fencing and (ii) 4 km of fencing, assuming that no fence is needed
along the rivers?
7 (c) Use implicit differentiation to find an equation for the tangent line to the graph of
sin(2x + y) = y
3
sin(x) at the point (0, 0).
Exam, Page 5 of 5
