AP Calculus Review Flashcards

Terms Definitions
Reimann Sums
dy/dx arctan(x)
(ln x)'
d/dx (c(f(x))
implicit equation
Mean Value Theorem
integral of a^xdx
Formula for Growth
lim h-->0 tan(x)/x=
d/dx (ln x)
degree 30⁰ rad=?
Average rate of change
When f'(x)>0
f(x) is increasing
lim h-->0 sin(x)/x= ?
(csc x)'
-(csc x)(cot x)
tan x
1/ cot x
backward difference quotient p.85
sin (A-B)=
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
the absolute value of velocity
the slope of a horizontal line
Integral of tan(x) dx
ln lsec(x)l +C
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
the possible y-values of a function
odd function
satisfies the property f(-x) = -f(x)
Integral of sec (x) dx
ln lsec(x)+tan(x)l +C
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.
The number that a function is approaching as x approaches a particular value from both the right and the left.
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
The derivative of a velocity function with respect to time
A function is continuous on an interval if and only if it is continuous at each point of the interval
amplitude p.113
the y-distance between the sidusoidal axis and a high point or a low point
[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/(b-a) * S(a, b) ƒ(x) dx
left-hand limit
The number that a function is approaching as x approaches a particular value from the left.
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.
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
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/(b-a) integral from b to a
The second definition of a Derivative
lim x--> a
f(x)- f(a)/ (x-a)
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 y-distance from the x-axis 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] / (b-a)
even Functions
A function y = f(x) is called an even function if f(- x) = f(x). Even functions are symmetrical about the y-axis.
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:
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/(b-a) ∫ (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 ]
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]/ (b-a)
What the derivative indicates about a monotonic function
If f'(x) > 0, then f is increasing. If f'(x) < 0 then f is decreasing
initial position
Trapezoidal Rule
dy/dx csc(x)
(arcsin x)'
-tan x
riemman sum
integral of csc^2(x)dx
integral of sin(x)dx
Formula for Decay
(tan x)'
(sec x)^2
cubic function
f(u)= u^3
degree 45⁰ rad=?
degree 0⁰ rad=?
Linearization of a function
integral of e^x dx
Formula for Half life
csc x
1/ sin x
sec x
1/ cos x
quotient rule
lodhi-hidlo/ lo squared
symmetric difference quotient p.85
Net area under a curve
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)
Integral of cot(x) dx
ln lsin(x)l +C
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
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
the method of approximating a definite integral over an interval using the function values at the left-hand endpoints of the subintervals determined by a partition
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 &amp; 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
vertical asymptote
When the y value increases or decreases without bound as the x value approaches a number.
An asymptote parallel to the y-axis.
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 -
A line segment between 2 points on a curve.
product rule
(fg)' = f ' g + g' f
An extremum is either a maximum or a minimum. Extrema is the plural of extremum.
∫ tan x dx (1)
ln |sec x| + c
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)] /(b-a) = 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^2-r^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.
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 y-value 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
One-Sided and Two-Sided 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/(a-b))*∫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
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