AP Calculus A Flashcards

Terms Definitions
(d/dx)uv
uv'+u'v
Distance
s(t)
tan(pi/2)
Undefined
cos(0)
1
d/dx[f(x)±g(x)]
f'(x)±g'(x)
a(0)
initial acceleration
Total Displacement
a>bv(t)dt
dy/dx cot(x)
-csc(x)^2
(arctan x)'
1/(1+x)
d/dx (e^x)
e^x
integral of sec(x)tan(x)dx
sec(x)+C
Volume of crossection
a>bA(x)dx
integral of csc(x)cot(x)dx
-csc(x)+C
dy/dx arcsin(x)
1/ sqrt(1-x^2)
(cos x)'
-sin x
sin²θ + cos²θ=
1
degree 60⁰ rad=?
π/3
a(t)
acceleration at any time
area between two functions
a>b[f(x)-g(x)]dx
When f'(x)<0
f(x) is decreasing
(sec x)'
(sec x)(tan x)
d/dx (f/g)
(gf' - fg')/g^2
trigonometric function
y=sinx, y=cosx, y=tanx
cos (A+B)=
cosAcosB - sinAsinB
s(t) = 0
object at origin
Total distance traveled
∫ |v(t)| dt
relative maximum
See Also:
local maximum
sin^2 x + cos^2 x
1
∫ e^x ds
e^x + c
optimization
in application, maximizing or minimizing some aspect of the system being modeled.
30-60-90 triangle; side=?
hyp=2; longside=√3; shortside=1
The solid lies between planes perpendicular to the x-axis at x=0 and x=4. The cross sections perpendicular to the axis on the interval [ 0 , 4 ] are square whose diagonals lie between y=-√x and y=√x. Find the volume.
16
speed
The absolute value of the velocity.
cos 2x (3)
1 - 2sin^2 x
average velocity
Displacement (change in position) divided by time traveled
odd
the graph of an ____________ function is symmetric with respect to the origin
Differential used for estimating change of function
dy=f'(x)dx (or dy=f'(x)changex
Rolle's Theorem
Suppose f is a function, differentiable for all points in the open interval (a, b) and continuous for all points in the closed interval [a, b]. If f(a) = f(b) = 0, then there is at least one number c between a and b for which f ' (c) = 0.
numerically
A form of expressing a mathematical relationship by presenting raw data (numbers).
differentiable
A function is differentiable at a point if the derivative of the function exists at that point.
At that point, the function must be continuous, and it must not have a "corner" ("teetering tangent") or a vertical tangent.
∫ cos x dx
sin x + c
differential equation
an equation that can be derived
MRAM
the method of approximating a definite integral over an interval using the function values at the midpoints of the subintervals determined by a partition
algebraic function
a function whose dependent variable satisfies a polynomial relationship with one or more independent variables
take the derivative
if you need more in denominator
Criteria for differentiability of f(x)
continuous and local linearity
A circle is defined by the equation x²+y²=1. Find the equation for a cross section of squares with bases perpendicular to the xy plane
4(1-x²)
independent variable
The variable that represents the input of the function; the variable is acted upon by the function. (usually an "x" value, represented on the horizontal axis on a graph.)
right-hand limit
The number that a function is approaching as x approaches a particular value from the right.
Mean Value Theorem
Suppose f is a function, differentiable for all points in the open interval (a, b), and continuous for all points in the closed interval [a, b]. Then there is at least one point c between a and b for which the derivative of f(c) equals the average rate of change of the function.
concavity
Which way a curve is bowed or cupped.
critical point
Points where a function f(x) satisfies either f ' (x) = 0 or f ' (x) does not exist.
L'Hospital's Rule
limit f(x)/g(x) = lim of f'(x)/g'(x), if 0/0, ∞/∞ and then use LHR
x→a x→a
domain
For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real.
tolerance p.85
the x-increment, h
(look at example on p.85)
sinusoid
where A,B,C and D stand for constant and the function can be either cosine or sine.
GENERAL EQUATION: f(x)=C+AcosB(x-D)
tangent line
a line that intersects a curve once and only once
Find the volume of the solid generated by revolving the region bounded by y = x^(½) and the lines y = 2 and x = 0 around the line x = 4
224/15 pi
rates of change
See Also:
average rate of change
instantaneous rate of change
parameter
A variable in an equation that is not the domain or the range variable.
For instance, in slope-intercept form of the equation for a line (y = mx + b), m and b are the parameters for slope and y-intercept, while x and y are the domain and range variables.
In the function y = 2x + 4, 2 and 4 are the values for the slope and y intercept parameters.
∫ tan x dx (2)
-ln |cos x| + c
piece-wise function
a function that is defined by applying different formulas to different parts of its domain
even function
Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f: f(x)= f(-x)
phase displacement p.113
the x-coordinate of the beginning of the "first" cycle, where the argument of the sine or the cosine equals zero.
sinusoidal axis p.113
the horizontal axis running along the middle of the graph
How do you find an AVERAGE rate of change of a function f(x) on the interval [a,b]?
f(b)-f(a)/b-a
4th to find anti D
area function (definite integral with variable upper limit)
rules for comparing functions
When using to compare magnitudes of polynomials, consider only the terms with the largest exponent. As x goes to infinity, a polynomial will usually outgrow a logarithm.
Exponential functions with bases greater than 1 will always outgrow polynomials and for any positive constants a, b > 1.
Exponential functions with bases greater than 1 will outgrow power functions, which will usually outgrow logarithmic functions.
instantaneous rate of change
f(x) has an instantaneous rate of change at "a" given by
∫ ln x dx
x ln x - x + c
axis of revolution
the line about which a solid of revolution is generated.
linear combination p.90
a sum of values each multiplied by some coefficient. (look at example p.90)
How do you find the derivative using the limit process?
f'(x)=lim f(x+a)-f(x)/a
x-a
concave up
A curve is concave up if it is bowed or cupped in such a way that it would hold water.
∫ sec x dx
ln |sec x + tan x| + c
second derivative test
if f'(c)=0 and f''(c)<0, then f has a local maximum at x=c. if f''(c)>0, then f has a local minimum at x=c
What is the process when differentiating implicitly?
1. Differentiate both sides with respect to x. (Remember to attach a y prime when differentiating y terms.)
2. Collect y prime terms on the left side of the equation.
3. Factor out y prime.
4. Solve for y prime.
odd functions
A function y = f (x) is called an odd function if f (- x) = - f (x). Odd functions are symmetrical about the origin.
area between two curves
If ƒ and g are continuous with ƒ(x) ≥ g(x) throughout [a,b], then the area between the curves y= f(x) and y=g(x) from a to b is the intergral of [f-g] from a to b
How do you find the displacement of an object when given a velocity function?
Integrate the velocity function.
Rev. Vol. Verti (y=f(x))
∫ from a to b of (2πrh) dx (r= distance from axis to x) + (h= top - bottom)
derivative with respect to x
, or the slope expressed as "rise over run."
First Derivative Test for Local Extrema
a. If f'(x) changes sign from negative to positive at x = c then f has a local minimum at x = c
b. If f'(x) changes sign from positive to negative at x = c then f has a local maximum at x = c
Volume - Measure from your axis of rotation
always measure (top - bottom) or (right - left)
Given y = x + 1 and y = x² . Find the volume of the cross sections formed by circles perpendicular to the base of the solid from [ 0 , 1 ]
1.073
definition of derivative p.81
( x or h form)
geometrical meaning: the slope of the line tangent to the graph at x=c.
derivative of a sum of two functions p.92
if f(x)=g(x)+h(x), where g and h are differentiable functions of x, then f'(x)=g'(x)+h'(x). Words: The derivative of a sum equals the sum of the derivatives. Differentiation distributes over addition.
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Term:
Definition:
Definition:

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