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High School 12
Math

FTC(2)

Hyesung Kang
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Added Question (10/ 20)
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Question 1

Multiple Choice

If g(x) = ∫[a to x] f(t) dt, what is g'(x)?

  • f(x)
  • f(a)
  • f'(x)
  • 0

Question 2

Multiple Choice

If f(x) = 3x^2, what is the derivative of g(x) = ∫[0 to x] f(t) dt?

  • 3x^2
  • 6x
  • x^3
  • 9x^2

Question 3

Multiple Choice

If f(x) is a constant function, what is the derivative of g(x) = ∫[a to x] f(t) dt?

  • f(x)
  • 0
  • c
  • f'(x)

Question 4

Multiple Choice

If a function is continuous on the interval [a, b], what does the Mean Value Theorem for Integrals guarantee?

  • The function has a maximum at x = a.
  • There exists a number c in [a, b] such that the integral is zero.
  • There exists a number c in [a, b] such that the area under the curve equals the area of a rectangle with height f(c).
  • The function is differentiable on (a, b).

Question 5

Multiple Choice

How do you find the average value of a function on the interval [a, b]?

  • By finding the maximum value of the function.
  • By dividing the integral of the function from a to b by (b-a).
  • By calculating the derivative of the function.
  • By finding the point of inflection of the function.

Question 6

Multiple Choice

What is the average value of f(x) = 2x on the interval [1, 4]?

  • 4
  • 5
  • 6
  • 8

Question 7

Multiple Choice

What is the purpose of the chain rule when evaluating integrals with variable upper limits?

  • To find the average value of the function.
  • To calculate the area under the curve.
  • To multiply by the derivative of the upper limit.
  • To simplify the function.

Question 8

Multiple Choice

Find the average value of f(x) = x² on the interval [0, 3].

  • 3
  • 4.5
  • 6
  • 9

Question 9

Multiple Choice

In the Fundamental Theorem of Calculus, what does the constant 'a' represent in the integral?

  • The upper limit of the integral.
  • The lower limit of the integral.
  • The average value of the function.
  • The derivative at x.

Question 10

Multiple Choice

What is the integral of f(x) = 3x from 0 to 5?

  • 25
  • 37.5
  • 50
  • 75
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