AP Calculus Review BIG IDEA #2: Derivatives & Motion

Question 1

If f'(a) = 0 and f''(a) = 0, then ...

Accepted Answer

no conclusion can be made

Question 2

d/dx ex =

Accepted Answer

ex ln e, ex

Question 3

If, at x = a, f is continuous and changes from increasing to decreasing, then ...

Accepted Answer

f has a local maximum at x = a

Question 4

Which of the following is a statement of the Chain Rule?

Accepted Answer

If u = g(x), then d/dx f(u) = f'(u) du/dx, d/dx f(g(x)) = f'(g(x)) g'(x), dy/du ⋅ du/dx = dy/dx, d(f(g)) = f'(g) dg

Question 5

d/dx xa =

Accepted Answer

axa-1

Question 6

d/dx ax =

Accepted Answer

ax ln a

Question 7

What does f'(a) represent?

Accepted Answer

The slope of the tangent line to f at x=a, The slope of f if you zoom in "infinitely far" at x=a, The limit of the slopes of secant lines through x=a, The instantaneous rate of change of f at x=a

Question 8

Which of the following implies that f(x) has a vertical tangent?

Accepted Answer

dx/dy = 0 at x = a, f is continuous at x=a and |f'(x)|→∞ as x→a

Question 9

d/dx f(g(x)) =

Accepted Answer

f'(g(x)) g'(x)

Question 10

If s is position, then velocity is ...

Accepted Answer

ds/dt

Question 11

If s is position, then speed is ...

Accepted Answer

|ds/dt|

Question 12

d/dx tan x =

Accepted Answer

sec2x

Question 13

The normal line to f at x = a is ...

Accepted Answer

y - f(a) = -1/f'(a) (x - a)

Question 14

d(f/g) =

Accepted Answer

g df - f dg / g²

Question 15

d/dx (f(x) ⋅ g(x)) =

Accepted Answer

f(x) g'(x) + g(x) f'(x)

Question 16

Which of the following differentiation properties is FALSE?

Accepted Answer

d/dx [f(x) ⋅ g(x)] = f'(x) ⋅ g'(x)

Question 17

d(fg) =

Accepted Answer

f dg + g df

Question 18

An object in motion along a line reverses direction when ...

Accepted Answer

the sign of the object's velocity changes

Question 19

If f'(a) = 0 and f''(a) < 0, then ...

Accepted Answer

f has a local maximum at x = a

Question 20

Which of the following is the average rate of change of f on [a,b]?

Accepted Answer

f(b) - f(a) / b - a

Question 21

d/dx ln x =

Accepted Answer

1/x ln e, 1/x

Question 22

d/dx sec x =

Accepted Answer

sec x tan x

Question 23

If f' < 0 on an interval, then...

Accepted Answer

f is decreasing on that interval

Question 24

d/dx cos x =

Accepted Answer

-sin x

Question 25

If f(a) is ≥ every other y-value of f in some open interval containing x = a, then ...

Accepted Answer

f(a) has a local (relative) maximum at x = a

Question 26

d/dx arctan x =

Accepted Answer

1 / 1 + x²

Question 27

d/dx arcsin x =

Accepted Answer

1 / √(1 - x²)

Question 28

d/dx arccot x =

Accepted Answer

-1 / 1 + x²

Question 29

If s is position and v is velocity, then acceleration is ...

Accepted Answer

dv/dt, d2s/dt2

Question 30

If f(a) is less than or equal to every other y-value of f, then ...

Accepted Answer

f(a) has a global (absolute) minimum at x = a

Question 31

If f'' > 0 on an interval, then...

Accepted Answer

f' is increasing on that interval, f is concave up on that interval

Question 32

A particle is slowing down if ...

Accepted Answer

Velocity and acceleration have different signs

Question 33

If the graph of f is concave down on [a,b], then the linear approximation to f at x = a ...

Accepted Answer

overestimates f(b)

Question 34

If f' is decreasing on an interval, then...

Accepted Answer

f'' is negative on that interval, f is concave down on that interval

Question 35

d( f(x) ) =

Accepted Answer

f'(x) dx

Question 36

A particle is speeding up if ...

Accepted Answer

Velocity and acceleration have the same sign

Question 37

d/dx logax =

Accepted Answer

1/x ln a

Question 38

d/dx π2 =

Accepted Answer

0

Question 39

d/dx arccos x =

Accepted Answer

-1 / √(1 - x²)

Question 40

If f(a) is ≤ every other y-value of f in some open interval containing x = a, then ...

Accepted Answer

f(a) has a local (relative) minimum at x = a

Question 41

Velocity is constant when...

Accepted Answer

acceleration is zero

Question 42

If f' is increasing on an interval, then...

Accepted Answer

f is concave up on that interval, f'' is positive on that interval

Question 43

d/dx f-1(x) =

Accepted Answer

1 / f '(y)

Question 44

Which of the following must be true if f'(a) = 0?

Accepted Answer

The graph of f has a horizontal tangent at x=a.

Question 45

d/dx [f(x) / g(x)] =

Accepted Answer

g(x) f'(x) - f(x) g'(x) / g(x)²

Question 46

An object is at rest when ...

Accepted Answer

v(t) = 0

Question 47

If the graph of f is concave up on [a,b], then the linear approximation to f at x = a ...

Accepted Answer

underestimates f(b)

Question 48

A point of inflection of f is where ...

Accepted Answer

f'' changes sign, f' changes direction

Question 49

d/dx cot x =

Accepted Answer

-csc2x

Question 50

d/dx csc x =

Accepted Answer

-csc x cot x

Question 51

A critical point of f ...

Accepted Answer

is a point in the domain of f where f' = 0 or does not exist

Question 52

If f'(a) = 0 and f''(a) > 0, then ...

Accepted Answer

f has a local minimum at x = a

Question 53

If f' > 0 on an interval, then...

Accepted Answer

f is increasing on that interval

Question 54

Which of the following is the linear approximation to f at x = a?

Accepted Answer

f(x) ≈ f(a) + f'(a)(x - a)

Question 55

If f(a) is greater than or equal to every other y-value of f, then ...

Accepted Answer

f(a) has a global (absolute) maximum at x = a

Question 56

If, at x = a, f is continuous and changes from decreasing to increasing, then ...

Accepted Answer

f has a local minimum at x = a

Question 57

If f'' < 0 on an interval, then...

Accepted Answer

f is concave down on that interval, f' is decreasing on that interval

Question 58

d/dx sin x =

Accepted Answer

cos x

AP Calculus Review BIG IDEA #2: Derivatives & Motion

Questions 58

AP Calculus Review BIG IDEA #2: Derivatives & Motion

Questions 58