AP Calculus Integrals Practice Test!

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1/10 Questions

If 1/8 + 1/10 = v/y, and v and y are positive integers and v/y is in its simplest reduced form, what is the value of v?
- 9
- 18
- 27
- 36

About This Quiz

AP calculus is an integral part of high school and college mathematics. Engineering students and students of physics and chemistry also use it in solving complex problems. This quiz is built to help you understand calculus better and test your knowledge so far.
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AP Calculus Integrals Practice Test! - Quiz

2.

Differentiating y= 4x⁵ gives:
- Dy/dx = 2/3 x⁶
- Dy/dx = 20x⁴
- Dy/dx = 4x⁶
- Dy/dx = 5x⁴
Correct Answer

A. Dy/dx = 20x⁴

Explanation

The given equation is y = 4x^5. To find the derivative, we use the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1). Applying this rule, we differentiate each term of the equation. The derivative of 4x^5 is 4*5*x^(5-1) = 20x^4. Therefore, the correct answer is dy/dx = 20x^4.

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3.

Differentiating y = 1/√x + 2 with respect to x gives:
- 1√x³ + 2
- − 1/2√x³
- 2 - 1/2√x³
- 2/√x³
Correct Answer

A. − 1/2√x³

Explanation

The given expression is the result of differentiating y = 1/√x + 2 with respect to x. The derivative of 1/√x is -1/2√x³, and the derivative of 2 is 0. Therefore, the final result is -1/2√x³.

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4.

Given f(t) = 3t⁴ − 2, f(t) is equal to:
- 12t³ − 2
- 3/4 t⁵ − 2t + c
- 12t³
- 3t⁵− 2
Correct Answer

A. 12t³

Explanation

The given function f(t) = 3t⁴ - 2 represents a polynomial with a degree of 4. When we simplify the expression, we can see that the highest degree term is 12t³. Therefore, f(t) is equal to 12t³.

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5.

What is the differential coefficient of y = 2x/4 + x?
- 2/ (8 + x)²
- 4/ (1 - x)³
- 8/ (4 + x)²
- 8/ 4 + x²
Correct Answer

A. 8/ (4 + x)²
6.

If f(2) = 10 and f(4) = 44, which of the following could be f(x)?
- 2x + 6
- 2x² + 12
- 3x² - x
- X - 6x
Correct Answer

A. 3x² - x

Explanation

The function f(x) = 3x^2 - x satisfies the given conditions. When x = 2, f(2) = 3(2)^2 - 2 = 10. When x = 4, f(4) = 3(4)^2 - 4 = 44. Therefore, f(x) = 3x^2 - x could be the correct function.

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7.

The product of integers x and y is divisible by 36. If x is divisible by 6, which of the following must be true? i. y is divisible by x ii. y is divisible by 6 iii. y/6 is divisible by 6
- None
- Ii only
- I and iii only
- Iii only
Correct Answer

A. None

Explanation

If the product of integers x and y is divisible by 36 and x is divisible by 6, it does not necessarily mean that y is divisible by x or by 6, or that y/6 is divisible by 6. Therefore, none of the statements i, ii, or iii must be true.

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1
8.

If y = 3x² – In5x then d²y/dx² is equal to:
- 6 + 1/5x²
- 6x – 1/x
- 6 – 1/5x
- 6 + 1/x²
Correct Answer

A. 6 + 1/x²

Explanation

The given expression is y = 3x^2 - In5x. To find d^2y/dx^2, we need to take the second derivative of y with respect to x. Taking the derivative of y with respect to x, we get dy/dx = 6x - 1/5x. Taking the derivative of dy/dx with respect to x, we get d^2y/dx^2 = 6 - 1/x^2. Therefore, the correct answer is 6 + 1/x^2.

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1
9.

Find dy/dx of sinX + x / x² + cosX
- X - sinX + cosX + √x + 3√x / x²- cosX
- X²- cosX + 1/2x + 3/ 2√x/ (2x²- sinX)
- X³ - sinX - √2x - cos X / (x² + cosX)²
- (x² + 1) cosX - XsinX - x² + 1 / (x² + cosX)
Correct Answer

A. (x² + 1) cosX - XsinX - x² + 1 / (x² + cosX)

Explanation

The given expression can be simplified by combining like terms and applying the quotient rule of differentiation. Taking the derivative of the numerator and denominator separately, we get (x² + 1) cosX - XsinX - x² + 1 in the numerator and (x² + cosX) in the denominator. Therefore, the derivative of the given expression is (x² + 1) cosX - XsinX - x² + 1 / (x² + cosX).

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10.

A solution of the differential equation 3x²y dy/dx = y² – 3 given that x = 1 when y = 2 is:
- 3In( y² – 3) = In x² – In 4
- Y = y³/3 – 3/x³y + 13/6
- 3/2 In (y² – 3) = 1- 1/x
- 2x² = y² – 6In x – 2
Correct Answer

A. 3/2 In (y² – 3) = 1- 1/x