C890 Calculus I

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Free C890 Calculus I Questions

1.

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

  • 3

  • 9

  • 1

  • 4.5

Explanation

​​​​​​​


2.

​​​​​​​

  • 0

  • 1

  • 3

  • ∞

Explanation

​​​​​​​


3.

For implicit differentiation on x2 + y2 = 25, what is dy/dx?

  • ​−xy

  • y/x

  • −y/x​

  • x + y

Explanation

​​​​​​​


4.

Tangent line to y = ex at x = 0?

  • y = ex

  • y = 1

  • y = x + 1

  • y = x

Explanation

​​​​​​​


5.

Implicit: xy = 1, find dy/dx​.

  • −x/y

  • y/x

  • x/y

  • −y/x​

Explanation

​​​​​​​


6.

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

  • π/5

  • π/3

  • π

  • 2π/5

Explanation

​​​​​​​


7.

Implicit: 

  • ​​​​​​​

  • ​​​​​​​

  • ​​​​​​​

  • ​​​​​​​

Explanation

​​​​​​​


8.

Derivative of f(x) = tan⁡−1(3x)?

  • ​​​​​​​

  • ​​​​​​​

  • 3tan⁡−1(3x)

  • ​​​​​​​

Explanation

​​​​​​​


9.

A classroom integral: ∫₀¹ x² dx = ?

  • 1/3

  • 1

  • 0

  • 1/2

Explanation

Explanation:

To evaluate the definite integral ∫₀¹ x² dx, use the power rule for integration. The antiderivative of x² is (x³)/3. Substitute the upper and lower limits: [(1³)/3] − [(0³)/3] = 1/3 − 0 = 1/3. This represents the exact area under the curve y = x² between x = 0 and x = 1.

​​​​​​​Correct Answer:

1/3


10.

In teaching u-sub, ∫x ex² dx = ?

  • ex² + C

  • x ex + C

  • ½ ex² + C

  • ½ ex + C

Explanation

Explanation:

To evaluate ∫x ex² dx, we use the substitution method. Let u = x², then du = 2x dx, which implies x dx = ½ du. Substituting into the integral gives ∫ex² x dx = ½ ∫eu du = ½ eu + C. Replacing u back with x² gives ½ ex² + C as the final answer.

​​​​​​​Correct Answer:

½ ex² + C


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