AP Calculus: Full Mem. Quiz

RGAAHH I HATE CALLCUALS

Find zeros of f(x)

f(x) = 0

The intersection of f(x) and g(x)

f(x) = g(x) or f(x) - g(x) = 0

Vertical Asymptotes of f(x)

lim f(x) = pos/neg infinity

x—> a

What are the two forms that should never written when evaluating limits on FRQ’s.

0/0 and infinity/infinity

lim f(x)

x—>a

a) what should you do first? 

b) What if the numerator is infinity? 

c) What if the denominator is infinity?

d) What if the numerator is 0?

e) What if the denominator is 0?

a) Plug in a

b) Limit is infinity

c) Limit is 0

d) Limit is 0

e) Limit is infinity

Given some f(x) and some value x=a.

1. What will f(a) generate

2. What will f’(a) generate

3. What will f ”(a) generate

1. Y-value

2. Slope

3. Concavity

Show that f(x) is continuous at   x = a

lim f(x) = lim f(x) = lim f(a)

x—> a- x—a+

Horizontal Asymptotes

lim f(x) OR lim f(x)

x—> infinity x—> -infinity

Write how the conditions for when to use  L’Hospital would be written on the FRQ portion of the AP exam for the function h(x) while                              h(x) = f(x)/g(x) at a. 

Use the notation provided

lim f(x) = infinity or 0 + lim g(x) = 0 or infinity

x—>a x—>a

Definition of a derivative for f(x)

lim f(x+h) - f(x)/h

x—>h

The average rate of change of f(x) on [a,b]

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

The equation of the tangent line to f(x) at x = a

y - f(a) = f’(a)(x-a)

The equation of the normal line to f(x) at (a)

y - f(a) = -1/f’(a) (x-a)

How do you find the x-values of horizontal tangents to f , given f’(x) = g(x)/h(x)

g(x) = 0

How do you find the x-values of vertical tangents to f  , given f’(x) = g(x)/h(x)

h(x) = 0

Derivative of h(g(x))

h’(g(x)) * g’(x)

What are the conditions for the IVT? 

For function f(x) from x=c to x=d

Must be continuous on the interval [c,d] and f( c) cannot equal to f(d) and

Given a piecewise function, show it is differentiable at x = a where the function rule splits

lim f’(x) = lim f’(x)

x—> a+ x—>a-

The derivative of f(x) s(x)

(f’(x) x s(x)) + (f(x) x s’(x))

The derivative of t(x)/b(x)

t’(x) x b(x) - t(x) x b’(x) / b(x)²

How to find Critical values of f(x)

f’(x) = 0 and infinity

How do you show work for The interval(s) where f(x) is increasing/decreasing, using the f’(x)

Increasing: f'(x)>0

Decreasing: f'(x)<0

How do you find Inflection points of f(x)

f''(x) = 0 and ∞ and where concavity changes

Explain how to find the  absolute maximum or minimum of f (x) on [a,b]

1) Find critical points 

2) use 1st or 2nd derivative test

3) Compare the points by substituting  them back into the original function

4) and check endpoints

 Mean Value Theorem formula where point c is a < c < b for f(x)

f’(x) = f(b) - f(a) / b-a

Given the graph of f’(x), how do you find critical points?

When f’(x) = 0 or when f’(x) is undefined

Intervals where the slope of f (x) is increasing.

Determine where f’’(x) > 0

How do you Find a Maximum of the graph f(x) on [a,b] - explain using 1st and 2nd derivative test

Find where f’(x) = 0 and determine where f’(x) changes from + to - AND f’(x) critical points < 0

What are the Two conditions of Mean Value Theorem for the interval [a,b]

1. Continuous on a closed interval

   1. Differentiable on a closed interval

State the 3 steps to finding an implicit derivative

1. Differentiate with respect to y

2. Collect y’ to one side

   1. Isolate for y’

What is the relationship between continuity, differentiability, and limits?

Differentiability —> continuity —> limit exists —> left limit equals right limit

Find the average rate of change of f’(x) on [t₁,t₂]

f'(t₂) - f'(t₁) / t₂-t₁

When is the tangent line Over or under-approximation

Determine the concavity of the graph by using d²y/dx²  if d²y/dx² < 0    it is an over approximation and d²y/dx² > 0 it is an under approximation

Vertical tangents to a polar curve

-f(x)sinx + f'(x)cosx = 0

Given f’’(x), how do you determine the concavity of f(x)

Find where f′′(x)=0 or is undefined to identify possible points of inflection.

Test intervals around these points to determine the sign of f′′(x)

Conclude concavity: f(x) is concave up where f′′(x)>0 and concave down where f′′(x)<0

How many minutes on average do you have on the AP exam for the following

1. Non Calc MC

2. Calculator MC

3. FRQ

1. 2 mins

2. 3 mins

   1. 15 mins

sin(2x)=

2sin(x)cos(x)

cos(2x)=

1. 1 - 2sin²(x)

2. cos²(x) - sin²(x)

   1. 2cos²(x) - 1

What does an extremum (maximum or minimum) on a function correspond to on its derivative?

f ′(x) is either zero or undefined.

How can you find concavity on f'(x)

Increasing f’(x) = Concave up, decreasing 𝑓′(𝑥) = Concave down

When is the 2nd derivative inconclusive

When f’’( c ) = 0

When does a function have jump discontinuity

When the left-hand and right-hand limits are not equal to each other.

When does a function have a removable discontinuity at x=c?

When the limit exists as x approaches c, but c is either not defined or is another value.

When does a function have infinite discontinuity?

when the left-hand or right-hand limit (or both) approach infinity or negative infinity 

What do the 1st derivative and 2nd derivative test have in common

Find extrema and need critical point

What do the 1st derivative test  and the concavity test have in common

Checking test points, setting “something” equal to zero

Explain the steps of 2nd derivative test

Find critical point then sub into 2nd derivative, + then min/- then max

Explain the steps of  the concavity test

Find inflection points, test points