AP Calc AB Final Review Sheet

Flashcards for AP Calculus AB Final Review

Factor Cancellation Technique

Takes out the factor from top to bottom.

Conjugate Method

Rationalizes square roots to simplify.

LCM Method for Complex Fractions

Simplifies smaller fractions from bigger fraction.

Special Trig Limit

lim (sin x)/x = 1 as x approaches 0

Limit goes to infinity and equals a constant

Horizontal asymptote

Infinite limit

Vertical asymptote

Piecewise Function Continuity

Check for a constant value; otherwise, it is DNE (Does Not Exist).

Requirements for Continuity

f(a) is defined, limit as x approaches a of f(x) exists, and f(a) equals the limit as x approaches a of f(x).

Domain

If even [0, ∞), if odd all real #.

Degree less/degree more

Limit is 0

Degree equal/ degree equal

Ratios of the highest power coefficient.

Degree more/degree less

Limit doesn't exist.

Intermediate Value Theorem

If f(x) is continuous on [a,b], f(a) < y < f(b) or vice versa, then there exists at least one value c such that f(c) = y.

Melted Average Velocity

Over a period of time

Instantaneous Velocity

At a particular moment.

Secant Line

Passes 2 points.

Tangent Line

Passes one point (super close 2 points).

f'(a)

Limit as x approaches a of (f(x) - f(a))/(x-a).

Relative Minimum

Negative to positive

Relative Maximum

Positive to negative

Normal Line

Slope is the negative reciprocal, same P.O.T.

Tangent Line Equation

y - y1 = y'(x - x1) - P.O.T: (x1, y1), y': slope

Requirements of Differentiability

Continuous and defined, cannot be a corner, cusp, or vertical tangent.

Power Rule

nx^(n-1)

Sum/Difference Rule

f'(x) ± g'(x)

Derivative of e^x

e^x

Derivative of ln(x)

1/x

Product Rule

f(x) * g'(x) + g(x) * f'(x)

Quotient Rule

(Low dHigh - High dLow) / Low^2

Derivative of tan(x)

sec^2(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of cot(x)

-csc^2(x)

v(t) = velocity function

s'(t)

a(t) = acceleration

v'(t) = s''(t)

Chain Rule

f'(g(x)) * g'(x)

Implicit Differentiation

Differentiate w/ respect to x, solve for dy/dx.

Logarithmic Differentiation

Use ln on both sides to neutralize exponents

Limits

What height a function intends to reach when approaching an x-value.

Hole

Limit exists and is defined.

Vertical Asymptote

DNE - goes to infinity.

Discontinuous

Limit does not exist.

Oscillating behavior

Limit does not exist.

Direct Substitution

Plug it in!

Identity Rule

lim x = c as x approaches c.

Coefficient Rule

b * lim f(x) as x approaches c.

Sum or Difference Rule

lim f(x) ± lim g(x) as x approaches c.

Product Rule

lim f(x) * lim g(x) as x approaches c.

Quotient Rule

lim f(x) / lim g(x) as x approaches c.

Composite Function Rule

f(lim g(x)) as x approaches c.

Power Rule

(lim f(x))^n as x approaches c.

Inverse Derivative

f'(f^-1(a))

Derivative of arcsin(x)

1/√(1-x^2)

Derivative of arccos(x)

-1/√(1-x^2)

Derivative of arctan(x)

1/(1+x^2)

Jerk

The rate of change of the slope of the tangent.

Speed

Instantaneous velocity.

Displacement

Area under the curve.

Distance traveled

Sum of absolute values of the distances between turning points