MATH 2100 C958 Calculus I

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Free MATH 2100 C958 Calculus I Questions

1.

Area under y = 4 − x2 from -2 to 2?

  • 32/3

  • 8

  • 16

  • 16/3

Explanation

​​​​​​​


2.

Using trapezoidal rule approximate ​​​​​​​

  • 0.375

  • 0.333

  • 0.5

  • 0.25

Explanation

​​​​​​​


3.

Use second derivative test: f(x) = 2x3 − 3x2 + 1, at x = 1?

  • Local min

  • Local max

  • Inflection

  • No conclusion

Explanation

​​​​​​​


4.

Volume by disks: y = x2, rotate about x-axis from 0 to 1?

  • π/5

  • π/3

  • π

  • 2π/5

Explanation

​​​​​​​


5.

In teaching the Intermediate Value Theorem, a teacher considers f(x) = x3 - x - 2 on [1,2]. Does it guarantee a root?

  • Yes

  • No

  • Only if differentiable

  • Depends on limit

Explanation

Explanation

Polynomials are continuous on all real numbers. Evaluate the endpoints: f(1) = 1 - 1 - 2 = -2 and f(2) = 8 - 2 - 2 = 4. Since f(1) and f(2) have opposite signs and (f) is continuous on [1,2], the Intermediate Value Theorem ensures there exists c ∈ (1,2)) with f(c) = 0. Differentiability is not required—continuity on the closed interval suffices.

​​​​​​​Correct Answer

Yes


6.

∫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


7.

Using the washer method: region between y = x and y = x3, rotate about y-axis, volume?

  • π

  • π/3

  • 2π/3

  • π/2

Explanation

​​​​​​​


8.

Monotonicity: f(x) = e−x2, increasing where?

  • x < 0

  • x > 0

  • Never

  • Everywhere

Explanation

​​​​​​​


9.

Differential: estimate √25.1​ using f(x) = √x at x = 25.

  • 5.1

  • 5.02

  • 5.005

  • 5.01

Explanation

​​​​​​​


10.

Derivative of f(x) = cot⁡−1(x2)?

  • ​​​​​​​ ​​​​​​​ ​​​​​​​

  • ​​​​​​​

  • ​​​​​​​

  • ​​​​​​​

Explanation

​​​​​​​


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