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)

Back

y = log (base a) x, y' =

Front

y' = 1/(x lna)

Back

rate

Front

derivative

Back

y = e^x, y' =

Front

y' = e^x

Back

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

Front

concave up

Back

If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =

Front

g'(x) = f(x)

Back

y = tan⁻¹(x), y' =

Front

y' = 1/(1 + x²)

Back

y = x cos(x), state rule used to find derivative

Front

product rule

Back

Intermediate Value Theorem

Front

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.

Back

Alternate definition of derivative

Front

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

Back

Fundamental Theorem of Calculus

Front

∫ f(x) dx on interval a to b = F(b) - F(a)

Back

right riemann sum

Front

use rectangles with right-endpoints to evaluate integrals (estimate area)

Back

left riemann sum

Front

use rectangles with left-endpoints to evaluate integral (estimate area)

Back

y = a^x, y' =

Front

y' = a^x ln(a)

Back

y = sin⁻¹(x), y' =

Front

y' = 1/√(1 - x²)

Back

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

Front

point of inflection

Back

y = csc(x), y' =

Front

y' = -csc(x)cot(x)

Back

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

Front

increasing

Back

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

Front

critical points and endpoints

Back

trapezoidal rule

Front

use trapezoids to evaluate integrals (estimate area)

Back

y = cot⁻¹(x), y' =

Front

y' = -1/(1 + x²)

Back

If f '(x) = 0 and f"(x) > 0,

Front

f(x) has a relative minimum

Back

Product Rule

Front

uv' + vu'

Back

If f '(x) = 0 and f"(x) < 0,

Front

f(x) has a relative maximum

Back

area under a curve

Front

∫ f(x) dx integrate over interval a to b

Back

average value of f(x)

Front

= 1/(b-a) ∫ f(x) dx on interval a to b

Back

y = sec(x), y' =

Front

y' = sec(x)tan(x)

Back

y = cos²(3x)

Front

chain rule

Back

[(h1 + h2)/2]*base

Front

area of trapezoid

Back

y = sin(x), y' =

Front

y' = cos(x)

Back

y = ln(x), y' =

Front

y' = 1/x

Back

y = tan(x), y' =

Front

y' = sec²(x)

Back

y = cos⁻¹(x), y' =

Front

y' = -1/√(1 - x²)

Back

Formal definition of derivative

Front

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

Back

y = cot(x), y' =

Front

y' = -csc²(x)

Back

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

Front

concave down

Back

When is a function not differentiable

Front

corner, cusp, vertical tangent, discontinuity

Back

Average Rate of Change

Front

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

Back

Instantenous Rate of Change

Front

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

Back

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

Front

relative maximum

Back

Linearization

Front

use tangent line to approximate values of the function

Back

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

Front

decreasing

Back

Particle is moving to the left/down

Front

velocity is negative

Back

y = ln(x)/x², state rule used to find derivative

Front

quotient rule

Back

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

Front

relative minimum

Back

absolute value of velocity

Front

speed

Back

Quotient Rule

Front

(uv'-vu')/v²

Back

Chain Rule

Front

f '(g(x)) g'(x)

Back

Section 2

(17 cards)

To draw a slope field,

Front

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

Back

cos(pi/3)

Front

1/2

Back

cos(pi/4)

Front

Back

volume of solid of revolution - washer

Front

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

Back

cos(pi)

Front

-1

Back

cos(pi/2)

Front

0

Back

sin(pi/6)

Front

1/2

Back

sin(pi/2)

Front

1

Back

sin(pi/4)

Front

Back

given v(t) find total distance travelled

Front

∫ abs[v(t)] over interval a to b

Back

cos(0)

Front

1

Back

sin(pi/3)

Front

Back

sin(0)

Front

0

Back

sin(pi)

Front

0

Back

cos(pi/6)

Front

Back

area between two curves

Front

∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function

Back

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

Front

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