AP Calc AB Final Prep

d/dx (1)

0

d/dx$\left(x^{n}\right)$

$$nx^{n-1}$$

d/dx (sinx)

$$\cos x$$

d/dx (cosx)

$$-\sin x$$

d/dx (secx)

$$\sec x\tan x$$

d/dx (cscx)

$$-\csc x\cot x$$

d/dx (tanx)

$$sec^2x$$

d/dx (cotx)

$$-\csc^2x$$

d/dx$\left(e^{x}\right)$

$$e^{x}$$

d/dx$\left(a^{x}\right)$

$$a^{x}\ln a$$

d/dx$\left(\log_{a}x\right)$

$$\frac{1}{x\ln a}$$

d/dx (lnx)

$$\frac{1}{x}$$

d/dx (arcsinx) or d/dx$\left(\sin^{-1}x\right)$

$$\frac{1}{\sqrt{1-x^2}}$$

d/dx (arcsecx) or d/dx$\left(\sec^{-1}x\right)$

$$\frac{1}{\left|x\right|\sqrt{x^2-1}}$$

d/dx (arctanx) or d/dx$\left(\tan^{-1}x\right)$

$$\frac{1}{1+x^2}$$

$$\int_{}^{}\!1\,dx$$

$$x+c$$

$$\int_{}^{}\!n\,dx$$

$$nx+c$$

$$\int\!x^{n}\,dx$$

$$\frac{x^{n+1}}{n+1}+c$$

$$\int^{}\!\sin x\,dx$$

$$-\cos\left(x\right)+c$$

$$\int^{}\!\cos x\,dx$$

$$\sin\left(x\right)+c$$

$$\int_{}^{}\!\sec^2x\,dx$$

$$\tan\left(x\right)+c$$

$$\int^{}\!\csc^2x\,dx$$

$$-\cot\left(x\right)+c$$

$$\int^{}\!\sec x\cdot\tan x\,dx$$

$$\sec\left(x\right)+c$$

$$\int^{}\!\csc x\cdot\cot x\,dx$$

$$-\csc\left(x\right)+c$$

$$\int^{}\!e^{x}\,dx$$

$$e^{x}+c$$

$$\int^{}\!\frac{1}{x}\,dx$$

$$\ln\left|x\right|+c$$

$$\int^{}\!a^{x}\,dx$$

$$\frac{a^{x}}{\ln a}+c$$

$$\int^{}\!\frac{1}{\sqrt{1-x^2}}\,dx$$

$$\sin^{-1}\left(x\right)+c$$

$$\int\!\frac{1}{1+x^2}\,dx$$

$$\tan^{-1}\left(x\right)+c$$

$$\int^{}\!\frac{1}{x\sqrt{x^2-1}}\,dx$$

$$\sec^{-1}\left(x\right)+c$$

What is a Critical Point?

When

$f^{\prime}\left(x\right)=0$ or DNE

What is a Point of Inflection?

When f ‘‘(x) changes signs

What is the Intermediate Value Theorem?

If f(x) is continuous on a closed interval [a,b], then f(x) takes on every y-value between f(a) and f(b)

What is the Extreme Value Theorem?

If f(x) is continuous on [a,b], then an absolute max and an absolute min are guaranteed on the interval.

What is the Mean Value Theorem?

If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a c-value such that:

$$f^{\prime}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

What is Average Value?

The integral over the interval

What is an Average Rate of Change of f(x) from [a,b]?

$$\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

List the steps for solving a differentiable equation:

i. Separate Variables

ii. Integrate both sides

• Plus C on the ride side only

iii. Use initial condition to find C

iv. Solve for y

What are the two values you need to write an equation of a tangent line?

Point and Slope

What does the sign of a 1st derivative tell you about the function?

Increasing and Decreasing

What does the sign on a 2nd derivative tell you about the function?

Concavity

What are the 4 ways a derivative fails to exist?

i. Corner

ii. Cusp

iii. Vertical Tangent

iv. Discontinuity

Differentiate the following with respect to time:

$$x^2+y^2=z^2$$

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

Find the derivative of:

$$x^2y-2x=y$$

$$x^2y^{\prime}+y\left(2x\right)-2=y^{\prime}$$

$$x^2y^{\prime}-y^{\prime}=2-2xy$$

$$y^{\prime}=\frac{2-2xy}{x^2-1}$$

If f ‘(x) changes from negative to positive, f(x) has a…

minimum

If f ‘(x) changes from positive to negative, f(x) has a…

maximum

If f(x) is increasing, f ‘(x) is…

positive

If f(x) is decreasing, f ‘(x) is…

negative

Given s(t) as a positive function, what is the particles displacement from [a,b]?

$$S\left(b\right)-S\left(a\right)$$

Given s(t) as a positive function, how do you evaluate v(3)?

Find S ‘(3)

How do you determine if a particle is slowing down?

Velocity and acceleration have opposite signs

When is a particle at rest?

$$V\left(t\right)=0$$

What is the speed of a particle?

$$\left|V\left(t\right)\right|$$

How do you determine if a derivative is equal to zero?

Numerator equals 0

How do you determine if a derivative is undefined?

Denominator equals 0

What are the steps to find extrema on a closed interval?

i. Find all critical points

ii. Evaluate all endpoints and critical points on f(x)

iii. OR make a sign chart

What is the Fundamental Theorem of Calculus if f(x) is the antiderivative of F(x) from (a,b)?

$$\int_{a}^{b}\!F\left(x\right)\,dx=f\left(b\right)-f\left(a\right)$$

If the rate of change of y is proportional to y, what is the equation to use?

$$y=Ce^{kt}$$

What do you use when justifying your answers?

What is given to you

What is L’Hopital’s Rule?

When a limit equals:

$$\frac00$$

or

$$\frac{00}{00}$$

Take the derivative of the top and bottom separately and plug them in again.

What is the 3 step continuity test at some c - value on f(x)?

i.

$f\left(c\right)$ exists?

ii.

$\lim_{h\to c}f\left(x\right)$ exists?

iii.

$f\left(c\right)=\lim_{x\to c}f\left(x\right)$ ?

Differentiability implies Continuity; TRUE or FALSE?

True; not the other way around!

What is the limit definition of a derivative?

$$\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$

What is the alternate limit definition of a derivative of f(x) as some c - value?

$$\lim_{x\to c}\frac{f\left(x\right)-f\left(c\right)}{x-c}$$

What is the product rule for derivatives?

1st factor times the derivative of the 2nd plus the 2nd factor times the derivative of the 1st

What is the quotient rule for derivatives?

Bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared

What is the chain rule for derivatives?

Derivative of the outside, leave inside alone, times derivative of the inside

What is the general form or an equation of a tangent line?

$$y-y_1=m\left(x-x_1\right)$$

What is the comparison between slopes of normal lines?

Opposite Reciprocals

What do you know about derivatives of inverse?

Reciprocals evaluated at reversed coordinates

How do you find the new position of a particle?

Initial + Displacement

What is the average velocity given position s(t) from (a,b)?

$$\frac{s\left(b\right)-s\left(a\right)}{b-a}$$

What are the 2 ways to find a particle’s total distance?

Multiple integrals with subtracting the negative integral or

$$\int^{}\!\left|v\left(t\right)\right|\,dt$$

What is the first derivative test for extrema at some point c?

If f ‘(x) changes from (+) to (-) then f(x) has a max

If f ‘(x) changes from (-) to (+) then f(x) has a min

What is the second derivative test for extrema at some point c?

If f ‘(c) = 0 and f ‘‘(c) > 0 then f(c) is a min

If f ‘(c) = 0 and f ‘‘(c) < 0 then f(c) is a max

What is the trapezoid rule formula for equal heights?

$$\frac12h\left(b_1+2b_2+2b_3+\cdots+2b_{n-1}+b_{n}\right)$$

What does

dy/dx$\int_{a}^{x}\!f\left(t\right)\,dt$

equal?

f(x)

Cross Section Formula of a Square:

$$\left(x\right)^2$$

Cross Section Formula of a Rectangle whose height is triple the base:

$$3\left(x\right)^2$$

Cross Section Formula of a Semicircle whose radius is x:

$$\frac12\pi\left(x\right)^2$$

Cross Section Formula of a Semicircle whose diameter is x:

$$\frac12\pi\left(\frac{x}{2}\right)^2$$

Cross Section Formula of an Isosceles triangle whose base is one of the legs:

$$\frac12\left(x\right)^2$$

Cross Section Formula of an Isosceles triangle whose base is the hypotenuse:

$$\frac14\left(x\right)^2$$

Cross Section Formula of an Equilateral triangle:

$$\frac{\sqrt3}{4}\left(x\right)^2$$

What is the formula using a disc method? (r(x) is the radius)

$$\pi\int_{a}^{b}\!\left(r\left(x\right)\right)^2\,dx$$

What is the formula when using the washer method?

(r(x) = small radius R(x) = big radius)

$$\pi\int_{a}^{b}\!\left(R\left(x\right)\right)^2\,-\left(r\left(x\right)\right)^2dx$$

How do you find the area between 2 curves?

Top minus Bottom

Right minus Left