Unit 1 of 5
Study guide for CLEP CLEP Calculus — Unit 1: Limits and Continuity. Practice questions, key concepts, and exam tips.
280
Practice Questions
33
Flashcards
4
Key Topics
Try these 5 questions from this unit. Sign up for full access to all 280.
If a function f(x) is differentiable at x = a, what can be concluded about its continuity at x = a?
It is not continuous
It is continuous
It is not defined
The limit does not exist
The derivative is zero
Answer: B — It is continuous is correct because differentiability implies continuity, since the existence of a derivative at a point requires the function to have a well-defined limit at that point, which is the definition of continuity.
Find the slope of y = - + x - 1 at x = 3.
10
12
14
16
20
Answer: D — 16 is correct because the derivative of y = x^3 - 2x^2 + x - 1 is y' = 3x^2 - 4x + 1, and at x = 3, y' = 3*3^2 - 4*3 + 1 = 27 - 12 + 1 = 16.
Find the derivative of y = (x).
2tan(x)(x)
2tan(x)csc(x)
2sec(x)tan(x)
(x)
2csc(x)cot(x)
Answer: A — Write y = (tan x)^2 and apply the chain rule: dy/dx = 2·tan(x)·(derivative of tan x) = 2·tan(x)·sec^2(x), since the derivative of tan(x) is sec^2(x). The choice 2·sec(x)·tan(x) is the derivative of sec(x), not of tan^2(x), so it is the common trap here.
Find the limit of the function f(x) = ( - 1) / x as x approaches 0 using L'Hospital's Rule.
The limit is infinity
The limit is 0
The limit is undefined
The limit is 1
The limit is -1
Answer: D — Apply L'Hospital's Rule to find the limit. The derivative of the numerator is e^x, and the derivative of the denominator is 1. So, the limit is $e^{0}$ / 1 = 1.
What is the limit as x approaches infinity of ?
0
1
-1
e
Answer: B — 0 is correct because as x approaches infinity, e^(-x) approaches 0 since the negative exponent causes the function value to decrease towards 0.
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