Terms  Definitions 

cos(pi/3) 
1/2

tan(pi/6) 
sqrt(3)/3

sin(pi/4) 
sqrt(2)/2

(d/dx)arctanu 
1/(1+u^2)

(d/dx)e^u= 
e^u(du/dx)

tanθ= 
sinθ/cosθ

Reimann Sums 
nSUMOFk=1f(ck)*changex

dy/dx arctan(x) 
1/(1+x^2)

(ln x)' 
1/x

d/dx (c(f(x)) 
cf'(x)

implicit equation 
f(x,y)=0

Mean Value Theorem 
f'(c)=(f(b)f(a)/(ba)

integral of a^xdx 
(1/ln(a))a^x+C

Formula for Growth 
u=Ae^(rt)

lim h>0 tan(x)/x= 
1

d/dx (ln x) 
1/x

degree 30⁰ rad=? 
π/6

Average rate of change 
avgROC=f(b)f(a)/ba

When f'(x)>0 
f(x) is increasing

lim h>0 sin(x)/x= ? 
1

(csc x)' 
(csc x)(cot x)

tan x 
1/ cot x

backward difference quotient p.85 
y/h

sin (AB)= 
sinAcosB  cosAsinB

When f''(x)<0 
f(x) is concave down

dy/dx = ky 
y = C*e^(kx)

1 + tan^2 x 
sec^2 x

d/dx (cos x) 
 sin x

speed 
the absolute value of velocity

zero 
the slope of a horizontal line

Integral of tan(x) dx 
ln lsec(x)l +C

asymptote 
A straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases without limit.
average rate of change 
cos 2x (2) 
2cos^2 x  1

range 
the possible yvalues of a function

odd function 
satisfies the property f(x) = f(x)

Integral of sec (x) dx 
ln lsec(x)+tan(x)l +C

graphically 
A form of expressing a mathematical relationship with a plot line that shows the relationship between the input (domain) of a function and the output (range) of a function.

limit 
The number that a function is approaching as x approaches a particular value from both the right and the left.

function 
A correspondence that relates one set of values (the domain) to a second set (the range) such that there is one and only one value of the range for each value of the domain.

∫ sec^2 x dx 
tan x + c

acceleration 
The derivative of a velocity function with respect to time

continuous 
A function is continuous on an interval if and only if it is continuous at each point of the interval

amplitude p.113 
the ydistance between the sidusoidal axis and a high point or a low point

optimization 
[max & min] 1) write function (goal), 2) helping equation put it into 1 variable, 3) find Criticals by setting f =0 or undef, 4) answer what asked w/ units

intermediate value theorem 
continuous on closed interval takes every value between f(a) and f(b)

Average Value Theorem 
1/(ba) * S(a, b) ƒ(x) dx

lefthand limit 
The number that a function is approaching as x approaches a particular value from the left.

symmetry 
An object has symmetry if it looks exactly the same when seen from two (or more) different vantage points. A function has symmetry if it is identical with its own reflection in an axis or point of symmetry.

global minimum 
The smallest value that the function has over its domain. (Also called an "absolute minimum.")

∫ dx/(a^2 + x^2) 
1/a Arctan (x/a) + c

absolute minimum 
The function ƒ has an absolute minimum value ƒ(c) at a point c in it's domain D if and only if ƒ(x) ≥ ƒ(c) for all x in D

logarithmic function 
a function y=log₋x, which is the inverse of y=a^x

inside function 
x = inside function EX: f(x)=sin(x )

composite function p.107 
composite function of the inside function with respect to x. It is composed of two other functions. EX: f(x)=sin(x ) In this example
sin is the outside function and (x ) is the inside function. 
related rates 
[given rates and you're asked for one] 1)goal rate, 2) given rate, 3) find equation that link to goal and to given, 4) do implicit differentiation, 5) substitute what you are given 6) solve for goal & check

Find the volume of the solid generated by revolving the region bounded by y = x² and the lines y = 1 around the line y=1 
16/15 π

Net Change Theorem 
ƒ(a) = ƒ(b) + S(a,b) ƒ'(x) dx

dot notation 
"x with dot on top" means "the derivative of x with respect to t." This notation always means the derivative with respect to time. Dot notation is traditional in physics settings.

Direction 
If v(t) > 0 then moving in positive direction.
If v(t) < 0 then moving in the negative direction If v(t) = 0 then the object is at rest 
derivative 
the function f' whose value at x is lim h>0 (f(x+h)f(x))/h, provided the limit exists

derivative function f'(x) p.85 
a function whose independent variable is x and whose dependent variable is the value of the derivative

chain rule p.107 
the method for finding the derivative of a composite function, namely the derivative of the outside function with respect to the inside function, multiplied by the derivative of the inside function with respect to x.

What is velocity? 
The derivative of position or the antiderivative of acceleration.

Average value of a function 
1/(ba) integral from b to a

The second definition of a Derivative 
lim x> a
f(x) f(a)/ (xa) 
Extreme Value Theorem 
On a closed interval for a continuous function, there exist a maximum and minimum function value.
***In order to find extreme value, evaluate the function at all critical values and endpoints. 
tangent line 
A line that touches a curve at one point and has the same slope as the curve at that point.

∫ csc x cot x dx 
 csc x + c

vertical displacement p.113 
the ydistance from the xaxis to the sinusoidal axis

Mean Value Theorem 2 
If f(x) is continuous over [a,b], then E at least one point x0 where = f(x0) =[∫ from a to b of f(x)dx] / (ba)

even Functions 
A function y = f(x) is called an even function if f( x) = f(x). Even functions are symmetrical about the yaxis.

local linearity p.82 
a function is locally linear at x=c if the graph of the function looks more and more like the tangent line to the graph as one zooms in on the point (c,f(c))

3rd to find anti D 
sum up area under the given graph (between a & b , the add f(a))

inverse function 
The inverse of f is written f  1. The domain and range of f  1 are respectively the range and domain of f.
f : f  1(x) = y if and only if f(y) = x.f must be one to one for f  to exist. 
Properties: Velocity & Acceleration 
If x is the displacement of a moving object from a fixed plane (such as the ground), and + is time, then the following are true:
velocity=v=v'=dx/dt acceleration=a=dv/dt=x"=d x/dt 
MVT(1) the slope of the tangent = 
the slope of secant at least once

Mean Value Theorem of Integrals 
The average value of a function. If f is continuous on [a,b] then there exists c in (a,b) such that f(c) = 1/(ba) ∫ (from a to b) f(x)dx

Fundamental theorem of calculus 
If f is a continuous on [a,b], and F is any antiderivative of f on [a,b], then ∫f(x)dx= F(b)F(a)

Given y = x + 1 and y = x² . Find the volume of the cross sections formed by right triangles with a leg perpendicular to the base of the solid from [ 0 , 1 ] 
.683

What is a Derivative (in words) 
 An instantaneous rate of change at a point
 Local Linear approximation (slope) 
Average value of a function over (a,b) 
[∫ from a to b of f(x) dx]/ (ba)

What the derivative indicates about a monotonic function 
If f'(x) > 0, then f is increasing. If f'(x) < 0 then f is decreasing

cos(pi) 
1

(d/dx)u/v 
vu'uv'/v^2

tan(pi/4) 
1

sin(pi/3) 
sqrt(3)/2

sin(pi/6) 
1/2

(d/dx)cotu 
csc^2(du/dx)

(d/dx)cosu 
sinu(du/dx)

(ax^n)' 
n*ax^(n1)

s(0) 
initial position

Trapezoidal Rule 
A=(h/2)(y+2y1+2y2...2yn1+yn)

dy/dx csc(x) 
csc(x)cot(x)

(arcsin x)' 
1/√(1x)

tan(x) 
tan x

riemman sum 
...

integral of csc^2(x)dx 
cot(x)+C

integral of sin(x)dx 
cos(x)+C

Formula for Decay 
y=ysub0e^(kt)

(tan x)' 
(sec x)^2

cubic function 
f(u)= u^3

degree 45⁰ rad=? 
π/4

degree 0⁰ rad=? 
0

Linearization of a function 
L(x)~f(a)+f'(a)(xa)

integral of e^x dx 
e^x+C

Formula for Half life 
ln2/k

csc x 
1/ sin x

sec x 
1/ cos x

quotient rule 
lodhihidlo/ lo squared

symmetric difference quotient p.85 
y/(2h)

Net area under a curve 
a>bf(x)dx

relative minimum 
See Also:
local minimum 
1 + cot^2 x 
csc^2 x

d/dx (Arctan x) 
1/(1 + x^2)

relative maximum/minimum 
relative maximum/minimum is higher/lower than any other point in its immediate vinicity

total anything during time interval given rate of change, R(t) 
T=a>bR(t)dt

Integral of cot(x) dx 
ln lsin(x)l +C

velocity 
The rate of change of position.

d/dx (csc x) 
csc x cot x

What is acceleration? 
The derivative of velocity.

Eqn of tangent line at a point 
yy1=m(xx1)

implicit differentiation 
Finding the derivative of implicit equations by differentiating each term of the equation with respect to the independent variable.

piecewise function 
A function that has different equations that describe the value of the function over different parts of the domain.

cos 2x (1) 
cos^2 x  sin^2 x

LRAM 
the method of approximating a definite integral over an interval using the function values at the lefthand endpoints of the subintervals determined by a partition

concavity 
the graph of a differentiable function y=f(x) is concave up on an open interval I if y' is increasing on I, or concave down on an open interval I if y' is decreasing on I.

What is nonremovable discontinuity? Describe analytically & graphically. 
Analyticall....you cannot remove common factors.
Graphically....there is no way to make the function continuous (vertical asymptote, jump, oscilation.) 
Differential Equation 
(separation of variable methods) = 1) get x on one side, y on the other , 2) integrate both sides, 3) solve for C using initial condition, 4) solve for y (if possible)

A circle is defined by the equation x²+y²=1. Find the equation for a cross section of equilateral triangles with bases perpendicular to the xy plane 
√3(1x²)

vertical asymptote 
When the y value increases or decreases without bound as the x value approaches a number.
An asymptote parallel to the yaxis. 
average velocity 
The distance traveled divided by the time of travel.

∫ cot x dx 
ln sin x + c

exponential function 
f(x)= c^x where c is some constant.

chain rule 
if y=f(u) is differentiable at the point u=g(x), and g is differentiable at x, then the composite function (f o g)(x)=f(g(x)) is differentiable at x, and (f o g)'(x)=f'(g(x))*g'(x)

acceleration p.99 
the instantaneous rate of change of velocity

speed p.99 
the absolute value of velocity. Tells how fast an object is going without regard to its direction

Local max occures where 
f'(x) changes from + to 

secant 
A line segment between 2 points on a curve.

product rule 
(fg)' = f ' g + g' f

extrema 
An extremum is either a maximum or a minimum. Extrema is the plural of extremum.

∫ tan x dx (1) 
ln sec x + c

Continuity 
If limit from the left = limit from the right = f(a) (a real #), then f is continuous at x=a

related rates equation 
an equation involving two or more variables that are differentiable functions of time that can be used to find an equation relating the corresponding rates

power function p.90 
derivative of the power of functions.
If f(x)=x , the f'(x)=nx Restriction: the exponent n is a constant EX: f(x)=x 
difference quotient p.80 
the ratio (change in f(x)/change in x). The limit of a difference quotient as the change in x approaches zero is the derivative.

Mean Value Theorem 1 
If f(x) is differentiable over (a,b), then E at least one point x0 where = f'(x0) = [∫ from a to b of f(a)  f(b)] /(ba) = avg rate of change

Fundamental theorem of calculus part one 
derivative of integral of f(t)=f(x)

Volume of Washer 
Integral from a to b of pi(R^2r^2) dx

local maximum 
A value of a function that is larger than or equal to the function values to its left and right. (Also called a "relative maximum.")

solid of revolution 
a solid generated by revolving a plane region about a line in the plane

inflection point 
a point where the graph of a function has a tangent line and the concavity changes

definition of derivative (derivative at x=c form) 
the instantaneous rate of change of f(x) with respect to x at x=c.

Differentiability 
If f is continuous at x = a and lim f'(x) (from the left) = lim f'(x) (from the right), then f is differentiable at x = a

differential equations p.119 
it is possible to find an equation of y as a function of +. EX: dy/dt=9.8t

Fundamental theorem of calculus part two 
integral from a to b = F(b)F(a)

intermediate value theorem (for continuous functions) 
a function y=f(x) that is continous on a closed interval [a,b] takes on every yvalue between f(a) and f(b)

initial condition p.120 
a given value of x and f(x) used to find the constant of integration.

Rev. Vol. Horiz (x=f(y) 
∫ from a to b of (2πrh) dy (r= distance from axis to y) + (h= right  left)

criteria for continuity of f(x) at x = a 
f(a) exist limit exist f(a)=limit

OneSided and TwoSided Limits 
If limit from the left = limit of the right = L, a real number L, then limit from both sides = L

derivative of a constant times a function 
if f(x)=kg(x), where g is a differentiable function of x, then f'(x)=kg'(x) provided k is a constant. Words: The derivative of a constant times a function equals the constant times the derivative of a function.

mean value theorem for definite integrals 
if f is continuous on [a,b] then at some point, c in [a,b], f(c)= (1/(ab))*∫f(x)dx (with bounds a,b)

theorem for limits of rational functions 
If "f " is a rational function and "a" is in the domain of "f " then:
. 
Second Derivative Test for local Extrema 
a. If f'(c) = 0 and f"(c) > 0 then f has a local minimum at x = c
b. if f'(c) = 0 and f"(c) < 0 then f has a local maximum at x = c 
Leave a Comment ({[ getComments().length ]})
Comments ({[ getComments().length ]})
{[ comment.comment ]}