Chapter 5 Calculus

Meaning of an Integral

area under a curve

Derivative (meaning, geometric meaning, and formula)

IROC, limit of AROC, slope – predictor function, slope of tangent line to the curve,

Lim h→0 (f(x+h)-f(x))/h

Point-Slope formula

y-y0 = m(x-x0)

Cusp

A point on the graph at which the function is continuous but the derivative is discontinuous. A sharp point or abrupt change in direction.

Power Rule

y=x^n y’=nx^n-1

Product Rule

y=uv      y’=u’v+uv’ (Derivative of the first times the second, plus the first times the derivative of the second.)

Quotient Rule

y=u/v      y=(u’v+uv’)/v² (Derivative of the top times the bottom minus derivative of the bottom times the top all divided by the bottom squared.)

Chain Rule

Derivative of the outside function multiplied by the derivative of the inside function.

Derivatives of the six trigonometric functions

sin’(x)=cos(x)

cos’(x)=-sin(x)

tan’(x)=sec²(x)

sec’(x)=sec(x)tan(x)

csc’(x)=-csc(x)cot(x)

cot’(x)=-csc²(x)

Critical Points

A point on a graph where the derivative is either zero or undefined.

Point of Inflection

A point where a graph changes from concave up to concave down or vice versa. f ”(x) = 0

or f ”(x) is undefined.

Reciprocals of Zero and Infinity

1/0→infinity    1/infinity→0    1/negative infinity→0

Antiderivative

The process of finding the original function when given the derivative.

Indefinite Integration

Same as the antiderivative.

Fundamental Theorem of Calculus

suppose f is continuous on [a,b]…

Derivatives and integrals of ln(u), and e^u

If y = ln(u) then y’ = (1/u)du (The derivative of natural log is one over what is after the natural log times the derivative of what is after natural log.)

If y = e^u then y’ = (e^u)du (The derivative of e to a variable power is e to that power times the derivative of the exponent.)

(e^u du = e^u + c (The integral of e to a variable power is e to the power plus a constant.)

((1/u) du = ln |u| + c (The integral of one over u to the first power is natural log of the absolute value of u plus a constant.)

Derivatives and integral of a^u

and log(a)u

(a^u du = (1/lna)a^u + c (The integral of a number “a” to a variable power is 1/ln(a) times the original function plus a constant.)

If y = a^u then y’ = (lna)a^u du (The derivative of a number “a” to a variable power is ln(a) times the original function times the derivative of the exponent.)

y = log(a)u then y’ = (1/lna)(1/u)(du) (The derivative of a log base “a” is 1/ln(a) times one over what is after the log times the derivative of what is after the log.)