Terms  Definitions 

rate 
derivative

(d/dx)(sec(x)) 
sec(x)*tan(x)

Quotient Rule 
(uv'vu')/v²

area below xaxis is 
negative

y = cos²(3x) 
chain rule

slope of vertical line 
undefined

slope of horizontal line 
zero

Product Rule 
uv' + vu'

[(h1  h2)/2]*base 
area of trapezoid

Midpoint Rule 
f(a+ΔX(k+1/2)) = f((Xk+Xk+1)/2)
Total Area: Mn=ΔX(f((X0+X1)/2)+...f((Xn1+Xn)/2) = ΔXsum(f((Xk+Xk+1)/2),n1,k=0) 
methods of integration 
substitution, parts, partial fractions

converges absolutely 
alternating series converges and general term converges with another test

Integral of: (csc(x))^2 dx 
cot(x) + c

x = ? (Polar) 
x = r*cos(@)

integral test 
if integral converges, series converges

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

y = sec(x), y' = 
y' = sec(x)tan(x)

y = tan(x), y' = 
y' = sec²(x)

y = e^x, y' = 
y' = e^x

Squeeze Theorem: 
Integral (1/x) = divergent, Integral (1/x^2) = convergent. If less than the former it is convergent, if more than the latter it is divergent.

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

trapezoidal rule 
use trapezoids to evaluate integrals (estimate area)

y = cos(x), y' = 
y' = sin(x)

arc length 
pi INT(a to b) SQRT(1 +(dy/dx)^2)dx

Riemann Sums 
ΔXsum(f(Xk),n1,k=0), where Xk is any sample ppoint in the kth subinterval

right riemann sum 
use rectangles with rightendpoints to evaluate integrals (estimate area)

Particle is moving to the right/up 
velocity is positive

First Derivative 
Shows the rate of change over time

Integral of: csc(x) dx 
lncsc(x)  cot(x) + c

left riemann sum 
use rectangles with leftendpoints to evaluate integral (estimate area)

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

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

y = tan⁻¹(x), y' = 
y' = 1/(1 + x²)

6th degree Taylor Polynomial 
polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative

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

y = sin⁻¹(x), y' = 
y' = 1/√(1  x²)

l'Hospital's Rule 
lim (f(x)) = lim (f'(x)) = lim (f''(x)) ...

indeterminate forms 
0/0, ∞/∞, ∞*0, ∞  ∞, 1^∞, 0⁰, ∞⁰

y = cot⁻¹(x), y' = 
y' = 1/(1 + x²)

Which functions are continuous everywhere? 
Polynomial, Rational, functions made up of continuous functions
i.e. f + g , f  g , f * g, f/g ( if g dne 0) 
Implicit Differentiation 
1. Separate y and x terms
2. Differentiate each term of the equation with respect to x 3. Factor out (dy/dx) on the left side of the equation 4. Solve for (dy/dx) 
dP/dt = kP(M  P) 
logistic differential equation, M = carrying capacity

Formal definition of derivative 
limit as h approaches 0 of [f(a+h)f(a)]/h

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

alternating series test 
lim as n approaches zero of general term = 0 and terms decrease, series converges

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

ratio test 
lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges

Instantenous Rate of Change 
Slope of tangent line at a point, value of derivative at a point

use integration by parts when 
two different types of functions are multiplied

Theorem: If the series sigma a(n) from 1  infinite is convergent... 
lim a(n) = 0

given velocity vectors dx/dt and dy/dt, find speed 
√(dx/dt)² + (dy/dt)² not an integral!

Derivatives of Exponential and Logarithmic Functions 
Let u be a differentiable function.
(e^u)' = e^u (a^u)' = a^u lnu (du/dx) 
use partial fractions to integrate when 
integrand is a rational function with a factorable denominator

find interval of convergence 
use ratio test, set > 1 and solve absolute value equations, check endpoints

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

Sum & Difference Rules for derivatives 
You can add or subtract derivatives
(g +/ f)' = g' +/ f' 
Second Derivative Test for Relative Extrema 
If f'(c)=0 and f''(c)<0, then f(c) is a relative maximum
If f'(c)=0 and f''(c)> 0, then f(c) is a relative minimum If f'(c)=0 and f''(c)=0, then f(c) is a relative maximum 
average value of f(x) 
= 1/(ba) ∫ f(x) dx on interval 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

For geometric series: sigma (a*r^(n1)) from 1 to infinite... 
Convergent if r < 1 and sum is a/(1r)
Divergent if r >= 1 
If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = 
g'(x) = f(x)

Test for Increasing/Decreasing Functions 
If f'(x) < 0 on [a,b] , then f(x) is decreasing on [a,b]
If f'(x) > 0 on [a,b] , then f(x) is increasing on [a,b] If f'(x) = 0 on [a,b] , then f(x) is a constant on [a,b] 
Fundamental Theorem of Calculus 
∫ f(x) dx on interval a to b = F(b)  F(a)

First Derivative Test for Relative Extrema 
If f'(x) changes from pos to neg at x=c, then f has a relative maximum at c
If f'(x) changes from neg to pos at x=c, then f has a relative minimum at c 
given rate equation, R(t) and inital condition when t = a, R(t) = y₁ find final value when t = b 
y₁ + Δy = y
Δy = ∫ R(t) over interval a to b 
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

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