AP Calculus BC Formulas/Review

What are limits?

As x approaches some value, f(x) approaches a value

The limit of several functions combined is equivalent to…

the sum, product, or quotient of their individual limits, provided each limit exists.

Limit as x approaches 0 of sin(x)/x

1

Limit as x approaches 0 of x/sin(x)

1

Limit as x approaches 0 of (1-cos(x))/x

0

Limit as x approaches 0 of (cos(x)-1)/x

0

Squeeze Theorem

If g(x)<f(x)<h(x) and is lim g(x)=L and lim h(x)=L then lim f(x)=L

Finding limits of trig functions (sine and cosine)

Use Squeeze Theorem since these functions are bounded by -1 and 1

Formal Definition of Continuity

A function f(x) is continuous at a point c if the following three conditions are met: f(c) is defined, the limit of f(x) as x approaches c exists, and it is equal to f(c)

Formal Definition of Removable Discontinuities

A function f(x) has a removable discontinuity at a point c if: the limit of f(x) as x approaches c exists, but it is not equal to f(c)

If the limit of f(x) as x approaches ± infinity equals L then…

f has a horizontal asymptote y=L

Average Rate of Change

(f(b)-f(a))/(b-a)

Instantaneous Rate of Change

f’(c)

Mean Value Theorem Part 1

f’(c)=(f(b)-f(a))/(b-a)

Mean Value Theorem Part 2

f(c)=(Integral from a to b)/(b-a)

Rolles Theorem

If f is continuous on [a,b] and differentiable on (a,b) and f(a)=f(b), then there exists a c in (a,b) such that f'(c)=0.

Average Value of a Function

Integral from a to b/(b-a)

Intermediate Value Theorem

If a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at least once.

Extreme Value Theorem

If f(x) is continuous on the interval [a,b], then f(x) must have both an absolute minimum and maximum on that interval

Inverse Function Theorem

If a function is differentiable and its derivative is non-zero at a point, then the function has a locally defined inverse corresponding to the area near that point.

Definition of a Derivative

f’(x)=lim h → 0 [f(x+h) - f(x)] / h

Derivative of x^n

n*x^n-1

Derivative of ln(x)

1/x

Derivative of log b (x)

1/(x*ln(b))

Derivative of e^x

e^x

Derivative of b^x

b^x * ln(b)

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec²(x)

Derivative of cot(x)

-csc²(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of arcsin(x)

1/sqrt(1-x²)

Derivative of arccos(x)

-1/sqrt(1-x²)

Derivative of arctan(x)

1/(1+x²)

Derivative of arccot(x)

-1/(1+x²)

Derivative of arcsec(x)

1/(|x|sqrt(x²-1))

Derivative of arccsc(x)

-1/(|x|sqrt(x²-1))

Product Rule

d/dx (f*g)=f’g+fg’

Quotient Rule

d/dx (f/g) = (f’g-fg’)/g²

Chain Rule

d/dx f(g(x)) = f’(g(x))*g’(x)

L’Hopital’s Rule

If lim x→a of f(x)/g(x) is indeterminate (0/0 or infinity/infinity), then lim x→a of f(x)/g(x) equals lim x→a of f’(x)/g’(x)

Derivative of the Inverse Function f^(-1)(x)

1/f’(f^(-1)(x))

Integral of x^n

(x^(n+1))/(n+1) + C, n ≠ -1

Integral of 1/x

ln|x| + C

Integral of f(g(x))

Use u-substitution and let u=g(x)

Integral of f(x)*g’(x)

Use integration by parts, f(x)*g(x)-(integral of f'(x)*g(x))

First Fundamental Theorem

Integral from a to g(x) of f(t) equals f(g(x))*g’(x)

Second Fundamental Theorem

If f(x)=g’(x), then the integral from a to b of f(t) equals g(b)-g(a)

Disc Method for Volume of f(x)

pi*integral from a to b (f(x))² dx when f(x) is the distance between the axis of revolution and the outside of the solid

Washer Method for Volume

pi*integral from a to b ((R(x))² - (r(x))²) dx when R(x) and r(x) are the outer and inner radii respectively.

Cross Section Method for Volume

Volume calculated by integrating the area of cross sections perpendicular to an axis, typically expressed as integral from a to b of A(x) dx, where A(x) is the area of the cross section perpendicular to the x-axis

Displacement using Velocity

The integral of velocity over time

Total Distance Traveled using Velocity

The integral of the absolute value of velocity over time

Derivative (dP/dt) of P=M/(1+Ce^(-kt))

k/M*P(M-P)

Carrying Capacity of P=M/(1+Ce^(-kt))

M

Euler’s Method when Approximating y_n+1

y_n+step size*derivative at y

Displacement from a to b

The integral of velocity from a to b

Total Distance Travelled of a Particle from a to b

Integral from a to b of the absolute value of velocity

Carrying Capacity of a Logistic Equation is L in…

dy/dt=ky(1-y/L) or dy/dt=k/L*y(L-y)

Point of Inflection of a Logistic Equation

Halfway between the carrying capacity and the starting value (normally 0)

Solution to the Logistic Differential Equation dy/dt=ky(1-y/L)

y(t) = L/(1 + Ce^{-kt}), where L is the carrying capacity

Rectangular Arc Length

Integral from a to b of sqrt(1+(dy/dx)²)

Parametric Arc Length

Integral from a to b of sqrt((dx/dt)² + (dy/dt)²)

Speed for a Parametric Equation

sqrt((dx/dt)²+(dy/dt)²)

Total Distance Travelled of a Parametric Equation

The integral from a to b of speed, or the integral from a to b of sqrt((dx/dt)² + (dy/dt)²) dt.

First Derivative of a Parametric Equation

dy/dx=(dy/dt)/(dx/dt)

Second Derivative of Parametric Equation

d²y/dx² = (d/dt(dy/dx))/(dx/dt) (numerator is the derivative of the first derivative)

Polar Area Under a Curve

½ * integral from a to b of r² where r is the polar function

Polar Conversion: r²=

x²+y²

Polar Conversion for x

r*cos(theta) where r is the polar function

Polar Conversion of y

r*sin(theta) where r is the polar function

Polar Conversion of theta

arctan(y/x)

nth Term Test

Diverges if the limit of the series as n approaches infinity does not equal 0

Geometric Series test for ar^n

If abs(r)<1 it converges, if abs(r) is greater than or equal to 1 it diverges

Sum of the Geometric Series a*r^n

a*r^k/(1-r) where a*r^k is the first term

p-series test 1/(n^p)

If p>1, it converges, if p is less than or equal to 1, it diverges

Alternating Series test

If the terms are decreasing and the limit of the series equals 0, then it converges

Integral test (Series=f(x))

Converges if the integral from 1 to infinity of f(x) converges, diverges if the integral from 1 to infinity of f(x) diverges

Ratio test

If the limit as n approaches infinity of (the next term/the current term) < 1, it converges, If the limit as n approaches infinity of (the next term/the current term) > 1, it diverges, inconclusive if it equals 1

Direct Comparison Test for Convergence

A series with terms smaller than a known convergent series will converge

Direct Comparison test for Divergence

A series with terms larger than a known divergent series will diverge

Limit Comparison

If the limit as n approaches infinity of a_n/b_n is finite and positive, then both series converge or diverge

When to use the limit comparison test

“messy'“ algebraic series, compare to a p-series

Maclaurin Series of e^x

1+x+x²/2!+x³/3!…+x^n/n!

Maclaurin Series of sin(x)

x-x³/3!+x^5/5!-…+(-1)^n*x^(2n+1)/(2n+1)!

Maclaurin Series of cos(x)

1-x²/2!+x^4/4!-…+(-1)^n*x^(2n)/(2n)!

Maclaurin Series for 1/(1+x)

1-x+x²-x³+…+(-1)^n*x^n

Maclaurin Series for 1/(1-x)

1+x+x²+x³+x^4+…+x^n

General Taylor Series for f(x) about x=a

f(a)+f’(a)(x-a)+f”(a)(x-a)²/2!+f”’(a)(x-a)³/3!+…+f^(n)(a)(x-a)^n/n!

Alternating Series Error Bound

a_n+1 (next term)

Lagrange Error Bound

f^(n+1)(z)*(x-c)^(n-1)/(n+1)! (next term except the derivative value is the highest possible on the interval)

What type of series is a Lagrange Error Bound for?

Taylor