AP Calc AB formula review

We're cooked

What is the derivative of any constant? (such as 2 or 69)

What is the derivative of any constant? (such as 2 or 69)

0

Product Rule

When you find the derivative of 2 things multiplying.

Quotient Rule

When you find the derivative of 2 things dividing

Chain Rule

When you have to find the derivative of the outside and inside. May apply more than once.

Mean Value Theorem

The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

What is the derivative of sin(x)?

cos(x)

What is the derivative of cos(x)?

-sin(x)

What is the derivative of tan(x)?

sec²(x)

What is the derivative of ln(x)?

1/x

What is the derivative of e^x?

e^x

What is the derivative of a^x? a represents any number with an exponent that has a variable.

a^x times ln(a) times x

What is derivative of sin^-1(x) or arcsin(x)?

(1)/(1-(x)²)^(1/2)

What is derivative of cos^-1(x) or arccos(x)?

-(1)/(1-(x)²)^(1/2)

What is derivative of tan^-1(x) or arctan(x)?

(1)/(1+(x)²)

L’Hospital’s Rule

We can only apply the L'Hospital's rule if the direct substitution returns an indeterminate form, that means 0/0 or ±∞/±∞. Apply it by finding the derivative of both the numerator and denominator.

∫k f(x)dx k represents any number. like in ∫ 2 f(x)dx

k∫f(x)dx (for example ∫ 2 f(x)dx becomes 2∫f(x)dx) Basically, as long as k is just a number and doesn’t have a variable like x or y, you can pull it out of the integral by making sure it still multiplies or divides.

How do you solve derivatives? (not for trig functions like sin or cos)

Multiply by the exponent and subtract 1 from the exponent. Derivative can have MANY variations in what it is called. It can be prime, derivative, slope of the tangent line, it can be lim x→h (f(x+h)-f(x))/h , d/dx, or more.

How do you solve for integrals? (not for trig functions like sin or cos)

Add 1 to the exponent and divide by the new exponent. Integral may also be referred to as anti-derivative or area under the curve. If there are bounds you got to write out a form where you plug in x or y as the upper bound and subtract by the form where you plug in x or y as the lower bound. If there are no bounds (Indefinite integral), then add +C to represent any constant.

How can you swap bounds of an integral and keep it same to the original? (sorry this is worded poorly)

Also multiply by a -1, example is in the picture. The reason why is if you swap the bounds, you’ll get the opposite sign version of the original with the correct bounds. Multiplying it by -1 changes it back to original. This will rarely come up though.

∫(1/u)

ln|u| + C

∫cos(u) du

sin(u) + C

∫sin(u) du

-cos (u) + C

∫tan(u) du

ln|sec u| + C

∫sec² u du

tan(u) + C

∫e^u du

e^u + C

∫a^u du

(a^u)/(ln a) + C

∫(1/(a²-u²)^(1/2))

sin^-1 (u/a) + C

or'

arcsin (u/a) + C

∫(1/(a² + u²)) du

(1/a) tan^-1 (u/a) + C

First Fundamental Theorem of Calculus

The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable upper bound.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is the formal, more general statement of the preceding fact: if f is a continuous function and c is any constant, then A(x)=∫xcf(t)dt A ( x ) = ∫ c x f ( t ) d t is the unique antiderivative of f that satisfies A(c)=0.

Assuming an integral has 2 bounds that are the exact same, what does it equal?

It equals 0

(poorly worded, sorry) How do you split an integral with the bounds b and a into ones that include a point c in the middle?

What is the formula for Arc Length?

Second Derivative Test

a systematic method of finding the absolute maximum and absolute minimum value of a real-valued function defined on a closed or bounded interval.

Point of Inflection

Point(s) at which concavity changes. Concavity is when it opens up or down.

How do you solve for point of inflection?

Find 2nd derivative and set it equal to 0. Then solve for the x’s. Once you find the x’s plug each of them back into the original equation and find each y value of the points (not shown in the image).

Concavity

Integrand

What we are finding the integral of.

ex: ∫ x² dx

the integrand is x²

Disk Method

The disk method is used when the axis of revolution is the boundary of the plane region and the cross-sectional area is perpendicular to the axis of revolution. This method is used to find the volume by revolving the curve y=f(x) y = f ( x ) about x -axis and y -axis.

Washer Method

Almost the same as the disk method. But it is Outer radius squared - Inner radius squared in the integral. Outer radius is which ever function is further away from axis of revolution in the bounds. If the axis of revolution is not at an axis, then it’ll be the axis of revolution minus the radius for that specific one. for example, if it’s at x=2, then the radius will actually be 2-(whatever the radius was supposed to be).

Top tier explanation, Ik (Joke)

Shell Method

another way to write it is 2π∫rh dx

r is radius = (axis of revolution) - x or y

h = height = f(x)

shell method isn’t as important to remember.

How do you access the integral symbol a ti-84 calculator?

MATH 9 (click math button than 9)