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Level 545

Calculus AB


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Intermediate Value Theorem
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.
average rate of change
Slope of secant line between two points, use to estimate instantanous rate of change at a point.
Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
increasing
When f '(x) is positive, f(x) is
decreasing
When f '(x) is negative, f(x) is
Relative minimum
When f '(x) changes from negative to positive, f(x) has a
Relative maximum
When f '(x) changes fro positive to negative, f(x) has a
concave up
When f '(x) is increasing, f(x) is
concave down
When f '(x) is decreasing, f(x) is
point of inflection
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
corner, cusp, vertical tangent, discontinuity
When is a function not differentiable
Product Rule
uv' + vu'
(uv'-vu')/v²
Quotient Rule
Chain Rule
f '(g(x)) g'(x)
Product Rule
y = x cos(x), state rule used to find derivative
Quotient Rule
y = ln(x)/x², state rule used to find derivative
Chain Rule
y = cos²(3x)
velocity is positive
Particle is moving to the right/up
velocity is negative
Particle is moving to the left/down
Speed
absolute value of velocity
y' = cos(x)
y = sin(x), y' =
y' = -sin(x)
y = cos(x), y' =
y' = sec²(x)
y = tan(x), y' =
y' = -csc(x)cot(x)
y = csc(x), y' =
y' = sec(x)tan(x)
y = sec(x), y' =
y' = -csc²(x)
y = cot(x), y' =
y = sin⁻¹(x), y' =
y' = 1/√(1 - x²)
y = cos⁻¹(x), y' =
y' = -1/√(1 - x²)
y = tan⁻¹(x), y' =
y' = 1/(1 + x²)
y = cot⁻¹(x), y' =
y' = -1/(1 + x²)
y' = e^x
y = e^x, y' =
y' = a^x ln(a)
y = a^x, y' =
y' = 1/x
y = ln(x), y' =
y' = 1/(x lna)
y = log (base a) x, y' =
critical points and endpoints
To find absolute maximum on closed interval [a, b], you must consider...
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(x) has a relative minimum
If f '(x) = 0 and f"(x) > 0,
f(x) has a relative maximum
If f '(x) = 0 and f"(x) < 0,
Linearization
use tangent line to approximate values of the function
rate
derivative
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
Trapezoidal Rule
use trapezoids to evaluate integrals (estimate area)
[(h1 - h2)/2]*base
area of trapezoid
definite integral
has limits a & b, find antiderivative, F(b) - F(a)
indefinite integral
no limits, find antiderivative + C, use inital value to find C
area under a curve
∫ f(x) dx integrate over interval a to b
Positive
area above x-axis is
Negative
area below x-axis is
average value of f(x)
= 1/(b-a) ∫ f(x) dx on interval a to b
g'(x) = f(x)
If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
To find particular solution to differential equation, dy/dx = x/y
separate variables, integrate + C, use initial condition to find C, solve for y
To draw a slope field,
plug (x,y) coordinates into differential equation, draw short segments representing slope at each point
zero
slope of horizontal line
use substitution to integrate when
a function and it's derivative are in the integrand
given v(t) find total distance travelled
∫ abs[v(t)] over interval a to b
area between two curves
∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function
volume of solid of revolution - no washer
π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution
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
The limit of f(x) as x approaches e
The existence or nonexistence of f(x) at x=c has no bearing on the existence of what?
b
What is lim (x→c) b?
c
What is lim (x→c) x?
c''
What is lim (x→0) x''?
Does not exist
What is lim (x→0) √x
Indeterminate form
0/0 is called what?
1
lim (x→0) [sinx / x] =
0
lim (x→0) [(1-cosx) / x] =
0
lim (x→0) [x / (sinx)] =
When is a function continuous on the open interval (a,b)?
When f is continuous at each point in the interval (a,b)
When is f continuous everywhere?
When f is continuous on the entire real line (-∞,∞)
Removable and nonremovable
What are the two types of discontinuity?
Essential
What is another term for non-removable discontinuity?
Removable
If a discontinuity at x=c can be made continuous by appropriately defining or redefining f(c), then f has what type of discontinuity
Right
Is lim (x→c⁺) f(x) a limit from the right or left?
left
Is lim (x→c⁻) f(x) a limit from the right or left?
0
lim (x→0⁺) ⁿ√x =
What is [x]?
Greatest integer function
What is the Intermediate Value Theorem
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
What is an infinite limit?
A limit in which f(x) increases or decreases with bound
If f decreases without bound, what does the limit approach?
⁻∞
If f decreases without bound, what does the limit approach?