# 1Multiple Choice
If g(x) = ∫[a to x] f(t) dt, what is g'(x)?
f(x)
f(a)
f'(x)
0
FTC(2)
Hyesung Kang
Freestyle
High 12
Math
FTC(2)
1
27
1
High 12
Math
FTC(2)
1
27
1
10 questions
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public quiz
# 2Multiple 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
# 3Multiple 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)
# 4Multiple 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).
# 5Multiple 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.
# 6Multiple Choice
What is the average value of f(x) = 2x on the interval [1, 4]?
4
5
6
8
# 7Multiple 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.
# 8Multiple Choice
Find the average value of f(x) = x² on the interval [0, 3].
3
4.5
6
9
# 9Multiple 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.
# 10Multiple Choice
What is the integral of f(x) = 3x from 0 to 5?
25
37.5
50
75
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