MATH 2100 C958 Calculus I

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Free MATH 2100 C958 Calculus I Questions

1.

Average value of f(x) = x2 on [0,3]?

  • 3

  • 9

  • 1

  • 4.5

Explanation

​​​​​​​


2.

For optimization problems in calculus, the final step after finding critical points is usually:

  • Use the first or second derivative test to classify

  • Plug into the original function

  • Check endpoints only

  • Graph the function

Explanation

Explanation:

After finding the critical points of a function (where f'(x) = 0 or f'(x)) is undefined), the next step in optimization problems is to determine whether each critical point corresponds to a local maximum, local minimum, or neither. This is typically done using the first derivative test, which examines the sign changes of (f'(x)), or the second derivative test, which uses the concavity of the function at the critical point (f''(x)) to classify the point. This step ensures that the correct values are identified for optimization purposes.

​​​​​​​Correct Answer:

Use the first or second derivative test to classify


3.

Implicit: exy = y, find dy/dx​.

  • y/1−xy

  • −xy/1+y

  • yexy /1−xyexy

  • exy/y

Explanation

​​​​​​​


4.

Newton’s method: approximate √2​ starting at x0 = 1, first iteration?

  • 1.4

  • 1.4167

  • 1.25

  • 1.5

Explanation

​​​​​​​


5.

For the function f, is f continuous at x = 1?
​​​​​​​

  • Yes

  • No, left limit differs

  • No, right limit differs

  • No, function undefined

Explanation

​​​​​​​


6.

Implicit: xy = 1, find dy/dx​.

  • −x/y

  • y/x

  • x/y

  • −y/x​

Explanation

​​​​​​​


7.

A teacher uses epsilon-delta: for f(x) = 3x + 2, prove lim⁡x→1f(x) = 5. What δ for ε=0.1?

  • 0.1/3

  • 0.1

  • 0.3

  • 1

Explanation

​​​​​​​


8.

Concavity change: f(x) = x4− 4x3, inflection points?

  • x = 1

  • x = 3

  • x = 0, 2

  • x = 0, 3

Explanation

​​​​​​​


9.

∫ x cos x dx = ? (integration by parts)

  • xsin⁡ x + cos⁡ x + C

  • sin⁡x + C

  • xcos⁡x + C

  • xsin⁡ x − cos⁡ x + C

Explanation

​​​​​​​


10.

d/dx [In |sec x|] = ?

  • tan x

  • sec x

  • cot x

  • 1

Explanation

​​​​​​​


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