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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
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)
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
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