Identify all the necessary conditions to apply the Mean Value Theorem of derivatives for a function f(x) on an interval.

The function is continuous on the interval.

The function is differentiable on the interval.

The function is positive on the interval.

Calculus Basics

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A short quiz on the basics of Calculus I

Questions and Answers

1.

Find the derivative of y=2x^2-3x+4
- A.
  
  Y'=4x-3
- B.
  
  Y'=2x-3
- C.
  
  Y'=4x^2-3
Correct Answer
A. Y'=4x-3

Explanation
use the power rule on each of the three terms: d/dx[x^n] = n*x^(n-1)

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2.

Identify all the necessary conditions to apply the Mean Value Theorem of derivatives for a function f(x) on an interval.
- A.
  
  The function is one-to-one.
- B.
  
  The function is continuous on the interval.
- C.
  
  The function is differentiable on the interval.
- D.
  
  The interval contains 0.
- E.
  
  The function is positive on the interval.
Correct Answer(s)
B. The function is continuous on the interval.
C. The function is differentiable on the interval.

Explanation
The Mean Value Theorem states that if a function is continuous and differentiable on an interval then somewhere in the interval the instantaneous rate of change is equal to the average rate of change over the interval.

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3.

If the derivative of a function is 0 at a point, the function has a relative maximum or minimum at that point.
- A.
  
  True
- B.
  
  False
Correct Answer
B. False

Explanation
Just because the derivative is 0 does not guarantee a maximum/minimum point. It just indicates that the function is "level", i.e., is neither increasing nor decreasing instantaneously at that point.

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4.

If the limit of a function at a particular x-value is equal to the function value at that x-value the function, at that point, is

Correct Answer
continuous

Explanation
The definition of Continuity at a point is the existence and equality of the function value and the limit at the point.

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4
5.

The Fundamental Theorem of Calculus depends on the function being continuous on the interval of integration.
- A.
  
  True
- B.
  
  False
Correct Answer
A. True

Explanation
A function may integrable even when it is not continuous, but the Fundamental Theorem of Calculus only applies when the function is continuous.

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Current Version
Jan 28, 2013

Quiz Edited by
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Jul 23, 2011

Quiz Created by
Shyamkurup

Calculus Basics

Find the derivative of y=2x^2-3x+4

Identify all the necessary conditions to apply the Mean Value Theorem of derivatives for a function f(x) on an interval.

If the derivative of a function is 0 at a point, the function has a relative maximum or minimum at that point.

If the limit of a function at a particular x-value is equal to the function value at that x-value the function, at that point, is

The Fundamental Theorem of Calculus depends on the function being continuous on the interval of integration.