Calculus test

Intermediate Value Theorem

If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis.

Average Rate of Change

Slope of secant line between two points, use to estimate instantanous rate of change at a point.

Instantenous Rate of Change

Slope of tangent line at a point, value of derivative at a point.

Formal definition of derivative

limit as h approaches 0 of [f(a+h)-f(a)]/h

Alternate definition of derivative

limit as x approaches a of [f(x)-f(a)]/(x-a)

When f '(x) is positive, f(x) is

increasing

When f '(x) is negative, f(x) is

decreasing

When f '(x) changes from negative to positive, f(x) has a

relative minimum

When f '(x) changes fro positive to negative, f(x) has a

relative maximum

When f '(x) is increasing, f(x) is

concave up

When f '(x) is decreasing, f(x) is

concave down

When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a

point of inflection

When is a function not differentiable

corner, cusp, vertical tangent, discontinuity

Product Rule

uv’ + vu’

Quotient Rule

(uv'-vu')/v²