MATH 2100 C958 Calculus I

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Free MATH 2100 C958 Calculus I Questions

1.

Global max of f(x) = x ln⁡ x − x on (0,∞)?

  • At x = 1/e

  • At x = 1

  • At x = e

  • None

Explanation

​​​​​​​


2.

Evaluate lim⁡x→3 (x2−9)/(√x−√3)​ by rationalizing

  • 6

  • 3

  • 9

  • 12√3

Explanation

​​​​​​​


3.

Absolute max of f(x) = x3 − 3x on [−2,2]?

  • 0

  • 4

  • -2

  • 2

Explanation

​​​​​​​


4.

Implicit: xy = 1, find dy/dx​.

  • −x/y

  • y/x

  • x/y

  • −y/x​

Explanation

​​​​​​​


5.

  • 0

  • 1

  • ∞

  • Does not exist

Explanation

Explanation

The real-valued function f(x) = √x is defined only for x ≥ 0. Approaching 0 from the left means considering values x < 0, where f(x) is not defined in the real numbers. Because there are no function values for x < 0, the left-hand limit cannot be evaluated and therefore does not exist as a real limit.

Correct Answer

Does not exist


6.

A high school teacher uses L’Hôpital’s rule on limₓ→0 (sin x / x). What is the result?

  • 0

  • 1

  • ∞

  • Undefined

Explanation

Explanation:

The limit limₓ→0 (sin x / x) is of the indeterminate form 0/0, so L’Hôpital’s rule applies. Differentiate the numerator and denominator: the derivative of sin x is cos x, and the derivative of x is 1. Therefore, the limit becomes limₓ→0 cos x = cos 0 = 1.

​​​​​​​Correct Answer:

1


7.

∫sec2 x dx =?

  • tan⁡ x + C

  • sec⁡ x + C

  • −cot ⁡x + C

  • ln⁡∣sec⁡ x∣ + C

Explanation

Explanation:

The derivative of tan⁡ x is sec⁡2 x, so the antiderivative of sec⁡2 x is:

∫sec⁡2 x dx = tan⁡ x +C

​​​​​​​Correct Answer:

tan x + C


8.

Evaluate lim⁡x→−∞ (√(x2+1))/ x + 1​​.

  • 1

  • 0

  • -1

  • ∞

Explanation

​​​​​​​


9.

Improper integral: ​​​​​​​

  • ∞

  • 1

  • 0

  • 1/2

Explanation

​​​​​​​


10.

Using simpson's rule ​​​​​​​

  • 2

  • 1.9

  • 2.1

  • 0

Explanation

​​​​​​​


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