Name:
Exam 2
Show all your work to receive full credit for a problem. There are a total of 72
points on this test.
1. (6 pts each.) Compute the following derivatives:
(a) f(x) = sin(ex2+x−1)
(b) g(x) = √1 + e2x
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Exam 2: Calculus Derivatives, Limits, and Implicit Differentiation, Exams of Calculus
A calculus exam with problems on derivatives, limits, and implicit differentiation. Students are required to compute derivatives of given functions, evaluate limits, find derivatives of composite functions, and solve an implicit differentiation problem. Additionally, there is a question on identifying antiderivatives and a problem on minimizing the sum of areas of two squares.
Typology: Exams
2012/2013
1 / 6
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Name:
Exam 2
Show all your work to receive full credit for a problem. There are a total of 72
points on this test.
- (6 pts each.) Compute the following derivatives:
(a) f (x) = sin(e x^2 +x− 1 )
(b) g(x) =
1 + e^2 x
- (6 pts each.) Evaluate the following limits:
(a) lim x→ 0
sin(2x)
x
(b) lim x→ 0
sin(2x)
sin x
(c) lim x→ 0
sin 2 (2x)
x sin x
- (6 pts.) Suppose the function y(x) is defined implicitly by the equation x^2 cos y+sin y =
x and y(1) = 0. What is y′(1)?
- (6 pts.) Suppose we have the function
f (x) =
e^2 x
1 + e^2 x^
Which of the following functions are antiderivatives of f? (Circle all that apply.)
(a) P (x) = arctan(ex) + 25
(b) R(x) = ln(
1 + e^2 x) + ln 3
(c) G(x) = 1 2 ln(1 +^ e
2 x )
(d) H(x) = ln(1 + e^2 x)
- (12 pts.) A 100-in. piece of wire is divided into two pieces and each piece is bent into
a square. How should this be done in order to minimize the sum of the areas of the two squares?