MATH 2100 C958 Calculus I

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Ace Your Test with MATH 2100 C958 Calculus I Actual Questions and Solutions - Full Set

Free MATH 2100 C958 Calculus I Questions

1.

Using simpson's rule ​​​​​​​

  • 2

  • 1.9

  • 2.1

  • 0

Explanation

​​​​​​​


2.

Area between y = sin⁡ x and y = cos ⁡x from 0 to π/4?​​​​​​​

  • √2​−1

  • 1

  • √2/2

  • 0

Explanation

​​​​​​​​​​​​​​


3.

Higher derivative: 4th derivative of f(x) = e−x?

  • e−x

  • −e−x

  • ex

  • 0

Explanation

​​​​​​​


4.

A teacher demonstrates squeeze theorem: if x2 ≤  sin⁡x/x ≤ 1 near 0, what is lim⁡x→0sin⁡ x/x​?

  • 0

  • 1

  • ∞

  • -1

Explanation

Explanation:

By the squeeze theorem, if a function is "sandwiched" between two other functions that have the same limit at a point, then the function also approaches that limit. Here, as x → 0, x2→ 0 and the constant 1 is 1. The standard limit lim⁡ x→0 sin⁡x/x​ is well-known to be 1. Therefore, despite the inequality given, the actual limit is determined by the behavior of sin⁡x/x near 0.

​​​​​​​Correct Answer:

1


5.

Related rates: cone height decreases at 1 cm/s, radius fixed at 3 cm, volume rate when h = 4?

  • −12π cm3/s

  • −9π cm3/s

  • −36π cm3/s

  • −3π cm3/s

Explanation

​​​​​​​


6.

Derivative of f(x) = cot⁡−1(x2)?

  • ​​​​​​​ ​​​​​​​ ​​​​​​​

  • ​​​​​​​

  • ​​​​​​​

  • ​​​​​​​

Explanation

​​​​​​​


7.

A student asks: “Why do we need the chain rule?” The teacher responds it is used when:

  • Differentiating a composite function

  • Finding critical points

  • Evaluating definite integrals

  • Applying L’Hôpital’s rule

Explanation

Explanation:

The chain rule is a fundamental rule in calculus used to differentiate composite functions. If a function can be expressed as f(g(x)), where one function is nested inside another, the derivative is found by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function: d/dx [f`(g(x))] = f'(g(x)).g'(x). This allows us to handle more complex functions that cannot be differentiated directly using simple rules.

​​​​​​​Correct Answer:

Differentiating a composite function


8.

In a lesson on substitution: ∫ln⁡x/x dx = ?

  • (ln⁡x)2 + C

  • 1/2(ln⁡x)2 + C

  • ln⁡∣ln⁡x∣ + C

  • x ln⁡x + C

Explanation

​​​​​​​


9.

A teacher analyzes f(x) = x2− 4x + 3. Where is it increasing?

  • x < 1 or x > 3

  • 1 < x < 3

  • Everywhere

  • x > 2

Explanation

​​​​​​​


10.

Volume by shells: rotate y = x2 from 0 to 1 about y-axis?

  • 2π/5

  • π/5

  • 2π/3

  • π/2

Explanation

​​​​​​​


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