AP Calculus Integrals Practice Test!

1. 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

To find the value of v, we need to add 1/8 and 1/10. To do this, we need to find a common denominator, which in this case is 40. So, 1/8 can be written as 5/40 and 1/10 can be written as 4/40. Adding these fractions together gives us 9/40. Since v/y is in its simplest reduced form, v must be 9.

Explanation

To find the value of v, we need to add 1/8 and 1/10. To do this, we need to find a common denominator, which in this case is 40. So, 1/8 can be written as 5/40 and 1/10 can be written as 4/40. Adding these fractions together gives us 9/40. Since v/y is in its simplest reduced form, v must be 9.

2. Differentiating y= 4x⁵ gives:

Dy/dx = 2/3 x6

Dy/dx = 20x⁴

Dy/dx = 4x6

Dy/dx = 5x⁴

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.

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.

3. Given f(t) = 3t⁴ − 2, f(t) is equal to:

12t³ − 2

3/4 t5 − 2t + c

12t³

3t5− 2

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.

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.

4. 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

not-available-via-ai

Explanation

not-available-via-ai

5. Differentiating y = 1/√x + 2 with respect to x gives:

1√x³ + 2

− 1/2√x³

2 - 1/2√x³

2/√x³

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.

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.

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

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.

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.

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

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.

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.

8. 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

not-available-via-ai

Explanation

not-available-via-ai

9. If y = 3x² – In5x then d²y/dx² is equal to:

6 + 1/5x²

6x – 1/x

6 – 1/5x

6 + 1/x²

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.

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.

10. 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)

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).

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).