AP Calc Mem Quiz (1-3)

Log_b(A ⋅ C) =

Log_bA + Log_bC

LogbA^B =

BLog_bA

(x,y) for cos and sine

(cosθ,sinθ)

lne =

1

ln1 =

0

log10 =

1

log 1 =

0

log_n1 =

0

lne^x =

x

e^lnx =

x

tan =

sinθ/cosθ

cotθ =

cosθ/sinθ

xⁿ ⋅ x^m =

x^n+m

(x^m)ⁿ =

x^mn

x⁰ =

1

EX: sin π/3 =

√3/2

EX: cos 3π/4

-√2/2

EX: tan 11π/6 =

-1/√3 = -√3/3

EX: cot 5π/4 =

1

Pythagorean Identity #1

sin²θ + cos²θ = 1

Pythagorean Identity #2 (divide by sin²)

1 + cot²θ = csc²θ

Pythagorean Identity #3 (divide by cos²)

tan²θ + 1 = sec²θ

Double Angle Identity #1

sin2θ = 2sinθcosθ

Double Angle Identity #2

cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

Double Angle Identity #3

tan2θ = 2tanθ/1 - tan²θ

log_b(E/F)

log_bE - log_bF

secθ =

1/cosθ

cscθ=

1/sinθ

d/dx xⁿ

nx^n-1

d/dx c

0

d/dx lnx

1/x

d/dx x

1

d/dx e^x

e^x

d/dx a^x

ln(a) * a^x

d/dx sinx

cosx

d/dx cosx

-sinx

d/dx tanx

sec²x

d/dx cotx

-csc²x

d/dx secx

secxtanx

d/dx csc

-cscxcotx

d/dx arcsinx

1/√1-x²

d/dx arccosx

-1/√1-x²

d/dx arctanx

1/1+x²

d/dx arccot

-1/1+x²

d/dx arcsecx

1/|x|√x²-1

d/dx arccscx

-1/|x|√x²-1

Quotient Rule d/dx t(x)/b(x)

(b(x))(t’(x)) - (t(x))(b’(x)) / (b(x))2

Product Rule d/dx f(x)s(x)

f'(x) s(x) + s'(x) f(x)

How to find the vertical tangent line

1) use the denominator

2) set the equation equal to x

a)Substitute in a

b) The limit is infinity

c) The limit is 0

d) the limit is 0

e) the limit is infinity

Definition of a derivative for f(x)

lim h→ 0 f(x+h)-f(x)/ h

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

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

When does a function have jump discontinuity

if the left-hand and right-hand limits exist but are not equal to each other

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

if the limit of the function exists as x approaches c, but the function is either not defined at c or is defined at c with a different value

3 requirements to know there is an inverse function

1. if they tell you f(x) and g(x) are inverse

2. f(x) and f^-1(x)

3. f(g(x)) = x or g(f(x))

f(^{x value of f(x)}) =

y value of f(x)

g(^{y value of f(x)}) = 

x value of g(x)

g(^{x value of f(x)}) = 

y value of g(x)

The 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

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

What will f(a) generate

What will f’(a) generate

a) y value

b) slope

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

Differentiability→continuity→limit exists→left limit equals right limit

d/dx sin(u)

cos(u) u ́

d/dx cos(u)

-sin(u) u´

d/dx tan(u)

sec²(u) u´

d/dx sec(u)

sec(u)tan(u) u´

d/dx csc(u)

-csc(u)cot(u) u´

d/dx cot(u)

-csc²(u) u´

d/dx e^u

e^u u´

d/dx ln(u)

u´ /u

d/dx a^u

a^u ln(a) u'

d/dx sin^-1(u)

d/dx cos^-1(u)

d/dx tan^-1(u)

d/dx arccot (u)

What will f’’(a) generate?

Concavity

Derivative of h(g(x))

h’(g(x)) x g’(x)

Steps for implicit

1) derive with respect to x

2) collect y' to one side

3) isolate for y’

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

Find extrema

Need critical point

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

Checking tests points

Setting something equal to 0

Two forms that should never be written

0/0 and infinity/infinity both require L hopital rule

Conditions for L hopital

Lim x → a f(x) = 0/infinity

Lim x →a g(x) = 0/infinity

Critical numbers of f(x)

f’(x) = 0 or infinity

Show work for intervals that decreasing or increasing using f’(x)

F’(x) > 0

F’(x) < 0

How to find inflection points of f(x)

F’’(x) = 0 and infinity and where concavity changes

How to find maximum and minimum

1) find critical points

2) use 1st/2nd derivative test

3) plug back into original function

4) check endpoints

In graph of f’(x) where do you find critical points of f(x)

Where f’(x) = 0 or where it is undefined

Intervals where the slope of f(x) is increasing

F’’(x) > 0

How do you find a maximum of the graph f(x) explain with 1st and 2nd derivative test

Find where f’(x) = 0 and where f’(x) changes from positive to negative and f’’(x) < 0

How much time per question on AP exam

Mcq and frq

Non calc mc is 2 minutes, calc is 3 and frq is 15

How to find concavity on f’(x)

Increasing f’(x)

Concave up

Decreasing f’(x)

Concave Down

When is 2nd derivative test inconclusive

F’’(x) = 0

F^-1 (x)

1/ f’(g(x))

Steps of 2nd derviative test

Find critical points

Substitute into 2nd derivative

plus mean min and negative means max

Concavity tests

Inflection points

Test points

Conditions for MVT

a) f(x) is continuous on the interval [a, b]

b) f(x) is differentiable on the interval (a, b)

c) f(a) = f(b)

Reasoning for MVT

(state the conditions) therefore there exists at least 1 value in (a, b) such that

f’(n) = f(b) - f(a) / b - a

Given a graph of f’’(x) and f'(a) = 0 and f’’(x) > 0, what do you know about x = a on f(x) besides it's critical point

f(x) has a minimum at x = a

Given a graph f’’(x), what do zeroes tell you about the graph of f(x)

f(x) has inflection points

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

f’(x) touches x-axis or is undefined