Which of the following represents the average value of f\left(x\right) on the interval \left[a,b\right]?

\frac{1}{b-a}\int_a^bf\left(x\right)dx

Which of these is the derivative Quotient Rule?

\frac{g\cdot f'-f\cdot g'}{g^2}

Which of the following derivatives does not require the Chain Rule?

\frac{d}{dx}\left[\frac{1}{x^3}\right]

What is the product rule?

d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x)

Which of the following trig derivatives is wrong?

\frac{d\left(\sec x\right)}{dx}=-\sec x\tan x

If g(x) = integral of f(t)dt from a to x where a is a constant, then g'(x) = (deriv of x)f(x)

2nd Form of the Fundamental Theorem of Calculus

What is the Chain Rule?

f'(g(x) . g'(x), a way to determine the derivative of a composite function, dy/du . du/dx, all of these

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AP Calculus Theorems

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Theorems and rules to review for AP Calc Exam

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Questions 23

Answers

1
Derivative of Sine
-\sin x
\frac{1}{\sin x}
\cos x
\frac{1}{\cos x}
2
MEan value theorem~~~
Yes
No
Not sure
3
Mean Value Theorem
f'(a) is undefined
avg rate of change = inst. rate of ch; (f must be differ.)
average rate of change = 0
instantaneous rate of change = 0
4
derivative of arccosx
1/(sqrt(1-x2)
-1/(sqrt(1-x2)
1/((1-x2)
5
The picture shows _____?
definition of an integral
L'Hospital's Rule
definition of a derivative
Mean Value Theorem
6
Which of the following is not part of the definition of continuity?
lim x→c f(x) = f(c)
lim x→c = 0
f(c) exists.
lim x→c f(x) exists.
7
Mean Value Theorem
\int _b^af\left(c\right)dx
\int _b^af\left(x\right)dx
\int_a^bf\left(x\right)dx=f\left(c\right)\left(b-a\right)
\frac{1}{b-a}\int _a^bf\left(x\right)dx
8
Which of the following represents the average value of f\left(x\right) on the interval \left[a,b\right]?
\frac{f\left(b\right)-f\left(a\right)}{b-a}
\frac{1}{b-a}\int_a^bf\left(x\right)dx
f\left(\frac{a+b}{2}\right)
\frac{f\left(a\right)+f\left(b\right)}{2}
9
This is the formula for ____?
General Volume
Arc Length
the Washer Method
the Disk Method
10
Trig Derivatives & Chain Rule
B
C
A
D
11
Derivative of Cosine
-\sin x
\frac{1}{\sin x}
-\cos x
\sin x
12
Which of these is the derivative Quotient Rule?
f'\left(x\right)-g'\left(x\right)
f\left(x\right)g'\left(x\right)+g\left(x\right)f'\left(x\right)
nx^{n-1}
\frac{g\cdot f'-f\cdot g'}{g^2}
13
Which of the following derivatives does not require the Chain Rule?
\frac{d}{dx}\left[\frac{1}{x^3}\right]
\frac{d}{dx}\left[e^{3x}\right]
\frac{d}{dx}\left[\sqrt{3x-1}\right]
\frac{d}{dx}\left[\sin^2\left(x\right)\right]
14
Which theorem is this?
Mean Value Theorem
Fundamental Theorem of Calculus
Rolles Theorem
Extrema Value Theorem
15
derivative of arctanx
1/(1+x2)
1/(1-x2)
-1/(1-x2)
16
The fundamental theorem of calculus states that The integral on (a, b) of f(x) dx = ____?
F(b) + F(a)
F(a) - F(b)
F(b) - F(a)
F(a) + F(b)
17
Definition of Derivative
B
A
D
C
18
What is the product rule?
d/dx (g(x)/ h(x)) = (h(x) g'(x) - g(x) h'(x))/ h(x)2
d/dx f(g(x)) = f'(g(x)) g'(x)
(d/dx)/(d/dy) = dx?dy
d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x)
19
Which of the following trig derivatives is wrong?
\frac{d\left(\cos x\right)}{dx}=-\sin x
\frac{d\left(\csc x\right)}{dx}=-\csc x\cot x
\frac{d\left(\cot x\right)}{dx}=-\csc^2x
\frac{d\left(\sec x\right)}{dx}=-\sec x\tan x
20
Which theorem is this?
Extrema Value Theorem
Rolle's Theorem
Fundamental Theorem of Calculus
Mean Value Theorem
21
mean value theorem
f'(c)=[f(b)-f(a)]/(b-a)
not really a thing
integral of f(x) from a to b
22
If g(x) = integral of f(t)dt from a to x where a is a constant, then g'(x) = (deriv of x)f(x)
Mean Value Theorem
Fundamental Theorem of Calculus
2nd Form of the Fundamental Theorem of Calculus
Change of Base Property
23
What is the Chain Rule?
f'(g(x) . g'(x)
a way to determine the derivative of a composite function
dy/du . du/dx
all of these

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