Full AP Calc BC Mem quiz

Mr Eaton full list

When can you use L’Hospital’s Theorem?

When the limit is ±∞/±∞ or 0/0

What does the Intermediate Value Theorem say?

If f(x) is continuous on [a, b], then f(c) exists between f(a) and f(b) such that c is between a and b

What are the 7 limit properties?

What is the Squeeze Theorem?

limx→0 (sinx/x)

1

limx→0 (1−cosx/x)

0

What is the average rate of change formula?

f(b) − f(a) / b − a

(d/dx)(uv)

u’v+u'v’

(d/dx)(u/v)

(u’v-uv’)/v²

(d/dx)(uⁿ )

nu^n-1 u’

(d/dx)(lul)

u/(lul)u’

(d/dx)(lnu)

1/u u’

(d/dx)(e^u )

e^u u’

(d/dx)(log_a u)

1/(ln(a)u u’

(d/dx)(a^u )

a^u ln(a)u’

(d/dx)(sin(u))

cos(u)u’

(d/dx)(cos(u))

-sin(u)u’

(d/dx)(tan(u))

sec² (u) u’

(d/dx)(cot(u))

-csc² (u) u’

(d/dx)(sec(u))

tan(u)sec(u) u’

(d/dx)(csc(u))

-cot(u)csc(u) u’

(d/dx)(sin^-1 (u))

1/√1−u2 u′

(d/dx)(tan^-1 (u))

1/1+u² u’

What is the definition of a derivative? (Give both)

Derivative of an inverse rule using f(x):

f′^-1(x) = 1/ f′(f^-1(x))

What does the Mean Value Theorem say?

If f(x) is differentiable on (a, b) and continuous on [a, b]

then c exists between (a, b) such that

f′(c) = f(b) − f(a)/b − a

What is the linearization (linear approximation)

formula?

L(x) = f(a) + f′(a)(x − a)

What does the first derivative test say?

When f’(x) changes from negative to positive f(x) has a

relative minimum value at that point.

When f’(x) changes from positive to negative f(x) has a

relative maximum value at that point.

What does the second derivative test say?

When f′′(c) < 0 then f(c) is a relative maximum.

When f′′(c) > 0 then f(c) is a relative minimum.

When f′′(c) = 0 then the test is inconclusive.

What does the extreme value theorem say?

If f(x) is continuous on the closed interval [a, b] and f(c) is the absolute maximum or minimum value on [a, b], then c is either a critical point or a or b.

∫ uⁿ du

∫ 1/u du

∫ e^udu

∫ a^udu

∫ sin u du

-cos(u) +c

∫ cos u du

sin(u) +c

∫ sec u du

∫ csc u du

∫ tan u du

∫ cot u du

∫ sec2 u du

∫ csc2 u du

∫ sec u tan u du

∫ cscucotu du

-csc(u) +c

∫1/√a2−u2 du

∫ 1/a2+u2 du

What are the 6 integral properties?

What are the seven integration methods?

1. Fundamental Theorem of Calculus (FTC)

2. U-Substitution

3. Integration by Parts

4. Long Division

5. Partial Fractions

6. Completing the Square (rare)

7. Improper Integrals

∫uv’ dx

uv- ∫ u’ v dx

What is the definition of an integral? (Give both)

What does the FTC part 1 say?

What does the FTC part 2 say?

What is Euler’s method? Show a two-step method.

y(x₀) = y₀

y(x₁) ≈ y₁ = y₀ + h ∙ f′(x₀, y₀)

y(x₂) ≈ y₂ = y₁ + h ∙ f′(x₁, y₁)

What does a Logistic differential equation look like?

(dP/dt) = kP(A − P)

(dy/dx) = ky(A − y)

Show what separation of variables with the following

integral: (dy/dx) = f(y)g(x)

∫ (1/f(y)) dy = ∫ g(x)dx

How do you find the area between the two curves

f(x) and g(x) if f(x) > g(x)?

What is the average value formula?

What are the displacement and distance equations in

linear motion if given v(t)?

Displacement: ∫ v(t)dt

Distance: ∫|v(t)|dt

What are the first and second derivatives of the

position function s(t)?

s′(t) = v(t)

s′′(t) = a(t)

How do you find volume by perpendicular cross section?

If f(x) > g(x) > r then how do you use the washer method rotated around y = r?

If you have the geometric series below and it converges, then what is the sum?

Write the Maclaurin Series for e^x:

Write the Maclaurin Series for 1/(1−x):

What does the Limit Comparison Test say about

positive series?

What does the Ratio Test say about ∑ a_n?

What does the nth term test say? (2 ways)

What is a p-series and when does it converge?

∑ (1/n^p)

When p > 1

What is the harmonic series? Does it converge or

diverge?

∑ (1/n)

Diverges

What is true for the Comparison Test if 0 ≤ a_n ≤ b_n

for all n ≥ N? (2 parts)

i) If ∑ b_n converges, then ∑ a_n also converges.

ii) If ∑ a_n diverges, then ∑ b_n also diverges.

What does the Alternating Series Error Bound say for

the partial sum of an alternating series up to a_N?

error < |a_N+1|

1. Continuous

2. Positive

3. Decreasing

The Maclaurin Series for sin x is:

The Maclaurin Series for cos x is:

What is a Geometric series and when does it

converge?

What do power series look like?

What is the form of a Taylor polynomial for f(x)?

Give the first four terms.

What is the form of a Maclaurin polynomial for f(x)?

Give the first four terms.

What is the difference between a radius of

convergence and an interval of convergence?

Radius: |x − c| < R

Interval: c − R < x < c + R test endpoints to determine

if it is equal to as well.

What is the Lagrange Error Bound for the n^th degree

Taylor Polynomial?

Integral:

Derivative:

In the following series when does the alternating

series test say that this converges?

What type of convergence happens when ∑|a_n|

converges and ∑ a_n converges?

Absolute convergence

What type of convergence happens when ∑|a_n| diverges and ∑ a_n converges?

Conditional convergence

Write down the arc length formulas for the following

curves: Function, Parametric, Polar

What is the difference between speed in linear

motion and speed in planar motion?

In parametric equations (dy/dx) and (d²y/dx²)

Given x(t) and y(t) what are the following vectors?: Position, Velocity, Acceleration

〈x(t), y(t)〉

〈x′(t), y′(t)〉

〈x′′(t), y′′(t)〉

What are the equations for all the following variables

in polar coordinates?: x, y, r, tan θ

What is the formula for area swept out by the polar

curve?