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### 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.

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

point of inflection

### To find absolute maximum on closed interval [a, b], you must consider...

critical points and endpoints

### mean value theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)

f '(c) = [f(b) - f(a)]/(b - a)

### To find particular solution to differential equation, dy/dx = x/y

separate variables, integrate + C, use initial condition to find C, solve for y

### To draw a slope field,

plug (x,y) coordinates into differential equation, draw short segments representing slope at each point

### volume of solid with base in the plane and given cross-section

∫ A(x) dx over interval a to b, where A(x) is the area of the given cross-section in terms of x

### volume of solid of revolution - no washer

π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution

### volume of solid of revolution - washer

π ∫ R² - r² dx over interval a to b, where R = distance from outside curve to axis of revolution, r = distance from inside curve to axis of revolution

### L'Hopitals rule

use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit

### second derivative of parametrically defined curve

find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt

### given velocity vectors dx/dt and dy/dt, find total distance travelled

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

### area inside polar curve

1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta

### area inside one polar curve and outside another polar curve

1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.

### Definition of Continuity

A function is continuous if 1) f(c) is defined. 2) lim f(x) as x approaches c exists 3) lim f(x) as x approached c equals f(c)

### Fundamental Theorem of Calculus Part 2

if f(x) is continuous on an open interval I, then d/dx[∫f(t)dt] = f(x)

### Rolle's Theorem

If f(x) is continuous on the closed interval [a, b], differentiable on (a, b), and satisfies f(a) = f(b), then for some c in the interval (a, b), we have f'(c) = 0