Find dy/dx of sinX + x / x² + cosX

(x² + 1) cosX - XsinX - x² + 1 / (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

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

AP Calculus Integrals Practice Test!

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Quizzes Created: 810 | Total Attempts: 987,351

Questions: 10 | Attempts: 353 | Updated: Mar 20, 2023

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AP Calculus Integrals Practice Test! - 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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Questions and Answers

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?
- A.
  
  9
- B.
  
  18
- C.
  
  27
- D.
  
  36
Correct Answer
A. 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.

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2.

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

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
- A.
  
  None
- B.
  
  Ii only
- C.
  
  I and iii only
- D.
  
  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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4.

Find dy/dx of sinX + x / x² + cosX
- A.
  
  X - sinX + cosX + √x + 3√x / x²- cosX
- B.
  
  X²- cosX + 1/2x + 3/ 2√x/ (2x²- sinX)
- C.
  
  X³ - sinX - √2x - cos X / (x² + cosX)²
- D.
  
  (x² + 1) cosX - XsinX - x² + 1 / (x² + cosX)
Correct Answer
D. (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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5.

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

A solution of the differential equation 3x²y dy/dx = y² – 3 given that x = 1 when y = 2 is:
- A.
  
  3In( y² – 3) = In x² – In 4
- B.
  
  Y = y³/3 – 3/x³y + 13/6
- C.
  
  3/2 In (y² – 3) = 1- 1/x
- D.
  
  2x² = y² – 6In x – 2
Correct Answer
C. 3/2 In (y² – 3) = 1- 1/x
7.

Differentiating y = 1/√x + 2 with respect to x gives:
- A.
  
  1√x³ + 2
- B.
  
  − 1/2√x³
- C.
  
  2 - 1/2√x³
- D.
  
  2/√x³
Correct Answer
B. − 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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8.

Differentiating y= 4x⁵ gives:
- A.
  
  Dy/dx = 2/3 x⁶
- B.
  
  Dy/dx = 20x⁴
- C.
  
  Dy/dx = 4x⁶
- D.
  
  Dy/dx = 5x⁴
Correct Answer
B. 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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9.

Given f(t) = 3t⁴ − 2, f(t) is equal to:
- A.
  
  12t³ − 2
- B.
  
  3/4 t⁵ − 2t + c
- C.
  
  12t³
- D.
  
  3t⁵− 2
Correct Answer
C. 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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10.

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