AP Calculus AB Exam Review

Absolute/Relative Min/Max

Acceleration

• Derivative (or change) in Velocity over time.
• Second derivative of position.

Asymptotes

• Refer to horizontal and vertical asymptotes.

Average Rate of Change

• Change in y over change in x (rise over run).

Average Slope

• Same as average rate of change.

Average Value

• Know the formula and the difference between average value and average rate of change.

Chain Rule (and Reverse Chain Rule)

• Derivative of the outside, with the inside staying the same, times the derivative of the inside.

Concavity

• The rate of change of the rate of change (second derivative).

Continuous

• If $\lim_{x \to a} f(x) = f(a)$, then the function is continuous at $x = a$.

Cosine Derivative

• $-sin(x)$.
• Remember that $\cos(0) = 1$ and then decreases, so its derivative starts negative.

Cotangent Derivative

• $-csc(x)^2$

Critical Point (Particle at Rest)

• Whenever the derivative is zero, or if a graph is pointy and not differentiable.

Cosecant Derivative

• $-csc(x) \cdot cot(x)$

Decreasing

• A slope is negative, so its derivative is below the x-axis.
• Never say "the slope is decreasing," say the slope is negative.

Derivative

• The slope of a tangent line, which measures the rate something is changing at one instant in time.

Differentiable

• If a graph is changing smoothly, then its derivative exists and it is differentiable.

Differential Equations

• Equations involving derivatives, which can be used to find information.
• Separate variables to find the antiderivative.

Displacement

• Ending minus starting position.
• How far you end up from where you started.
• Integrate the velocity to find displacement.

e^stuff Derivative

• Derivative of the stuff times original $e^{stuff}$.

Equation of Tangent Line

• $y - f(a) = f'(a)(x - a)$, where a is some x-value given, f(a) is the y-value, and f'(a) is the derivative at a.

Extreme Value Theorem

First Fundamental Theorem of Calculus

• Area equals the change in the anti-derivative.

Graph of f' or g'

• The graph of a derivative can be used to find lots of things.

Holes

• Also called "removable discontinuities."

Horizontal Asymptote

• x is approaching infinity.

Implicit Differentiation

• When y is part of the equation, find the derivative using the appropriate rule, then multiply it by y'.

Implicit Differentiation (2nd Derivative)

• If you have to find the second derivative, do the same thing as implicit differentiation and replace y' with the first derivative.

Increasing

• A slope is positive, so its derivative is above the x-axis.

Instantaneous Rate of Change

• Another name for derivative.

Integral (Definite)

• Means two things: Area and Anti-derivative.

Integral Rules

• Know reverse power rule and reverse chain rule.

Intermediate Value Theorem

Inverse

L'Hopital's Rule

• Use for $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
• It's the derivative of the top over the derivative of the bottom.

Limits (from a Graph)

• If the left and right limits meet, the limit exists, even with a hole.

Limits (L'Hopital's)

• Plug in the x-value of the limit; if you get $\frac{0}{0}$ or $\frac{\infty}{\infty}$, use L'Hopital's Rule.

Ln(stuff) Derivative

• Derivative of inside stuff, times $\frac{1}{the inside stuff}$.

Mean Value Theorem

Over/Under Approximations

Point of Inflection

• Whenever $f''$ changes signs (use a sign chart or look at the graph of $f''$).
• OR whenever the slope of $f'$ changes from positive to negative or vice versa.

Position

• Measures where something is located.
• Does not tell you how fast something is moving or what direction it's moving.

Position, Velocity, Acceleration

Power Rule

• Know this procedure for finding a derivative.

Product Rule

• Know this procedure for finding a derivative.

Quotient Rule

• Know this procedure for finding a derivative.

Related Rates

• It's like implicit differentiation, but every variable gets a prime.
• You may recognize a problem like this if they give you a variable t, for time, but t is nowhere in the equation.

Relative Maximum

Relative Minimum

Reverse Chain Rule

• First thing to always do: divide by derivative of the inside!

Riemann Sums

• Know the formulas and when they over or underapproximate.

Secant Line

• Connects two points of a function.

Second Fundamental Theorem of Calculus

• The derivative cancels out an integral.

Secant Derivative

• $sec(x) \cdot tan(x)$

Sigma (Summation)

• See notes.

Sine Derivative

• $cos(x)$
• Remember sin(0) is 0 and then increases, so its derivative starts positive.

Slope Field

• Plug in coordinates to the differential equation to see what its slope would be at that spot.

Slope of Tangent Line

• This is another way of saying "derivative."

Speed

• The absolute value of velocity.
• Speed is always positive.

Summation

• See Sigma above.

Tangent Line

• It graces a function at one exact point.
• Equation: $y - f(a) = f'(a)(x - a)$, where a is some x-value given, f(a) is the y-value, and f'(a) is the derivative.

Tangent Derivative

• $sec(x)^2$

Theorems

Trigonometry

• Draw a unit circle if you have to.
• Remember that 180 degrees is pi, 90 is pi/2, cosine is the x coordinate and sine is the y.

Units

• For position there is no "per," it's just a basic unit like meters.
• For velocity, or first derivative, or rate, it will be "y per x."
• For acceleration, or rate of the rate, or second derivative, it's "y per x-squared."

Velocity

• Tells you the speed something is traveling and the direction (in Calc AB it's either left or right).
• Derivative of the position

Vertical Asymptote

• Typically this occurs when you divide by 0.
• $ln(x)$ also has a vertical asymptote at x = 0.