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ap calculus bc semester 1 review
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Gravity
Key Concepts:
Terms in this set (94)
area of rectangle
lw
area of circle
area of triangle
(1/2)(bh)
area of trapezoid
(1/2)(b1 + b2)(h)
volume of rec. solid
lwh
volume of sphere
volume of cylinder
volume of cone
surface area of cylinder
surface area of a box
optimization steps
1) identify all quantities and draw a sketch
2) primary equation and secondary equation (constraints)
3) reduce primary equation to only one variable by manipulating secondary equation
4) plug secondary equation into primary equation
5) find the derivative of that one single variable
6) get the variable and plug back in for other variable
7) make sure you answered the question in right units, etc.
tangent line approximation
1) find first derivative
2) after finding derivative, plug in x-value in order to find SLOPE to build equation
3) after finding the slope, build your equation using your given ORDERED PAIR
4) find second derivative to determine concavity at the X-VALUE
5) plug in your x-value of approximation into your equation in order to fine your approximation
if concavity is upwards
underestimate
if concavity is downwards
overestimate
when velocity and acceleration have same signs
speed is INCREASING
when velocity and acceleration have different signs
speed is DECREASING
when acceleration is 0
speed is CONSTANT
speed is the ______________ of velocity
ABSOLUTE VALUE
if velocity is positive and acceleration is positive
it's moving right and gaining speed
if velocity is positive but acceleration is negative
it's moving right and slowing down
if velocity is negative but acceleration is positive
it's moving left and slowing down
if velocity is negative but acceleration is negative
it's moving left and gaining speed
when velocity changes signs, the object
changes direction
when velocity is at 0
the object is at rest
how do you find total distance
sum of the absolute values of the differences in POSITION (s(t)) between all resting points (CRITICAL POINTS of VELOCITY...or you can just do all points)
slope fields
1) easy, just plug in values and draw slopes
2) if matching, draw the curve with y=...if it's a dy/dx, plug in values to test
implicit differentiation
1) differentiate both sides while putting dy/dx every time you differentiate y (e.g. x - y.... 1 - dy/dx)
2) move all dy/dx to left of equation by factoring out and moving everything else to the other side
3) solve for dy/dx by dividing
derivatives of inverse functions
d/dx [arcsinnu]
d/dx [arccosu]
d/dx [arctanu]
d/dx [arccotu]
d/dx [arcsecu]
d/dx [arccscu]
related rates
1) identify all quantities and aspects by writing out FIND, GIVEN, WHEN, AND EQUATION
2) implicitly differentiate all variables of the equation
3) plug in what's given
4) solve for what you tryna find
5) voila make sure your units are correct
IF YOU HAVE HIDDEN VALUES PLUG THEM IN B4 YOU IMPLICITLY DIFFERENTIATE
reasons for limits not existing
1) f(x) approaches a diff. number from the right side of the c than it approaches from the left side
2) f(x) increases or decreases without bound as x approaches c
3) f(x) oscillates between two fixed values as x approaches c
ways to evaluate limits analytically
if it yields an indeterminate)
1) factor
2) multiply by conjugate
3) rewrite expression using trig identity
4) simplify (add/subtract fractions, reduce fractions)
5) use special limits
three special limits 1)
three special limits 2)
three special limits 3)
limit as x approaches 0 = (1+x) raised to (1/x) = e
l'hopital's rule
only if indeterminate
limits at infinity rules (horizontal asymptotes)
how to check for continuity?
1) f(c) is defined
2) lim f(x) as x approaches c exists
3) lim f(x) as x approaches c = f(c)
formal definition of a derivative
replace with delta x
alternative form of a derivative
differentiability and continuity
differentiability implies continuity
equations of a tangent line to a curve
1) find derivative of equation
2) plug in x value into that equation
3) bam that's your slope
4) now use the point slope formula to find your tangent line to the curve
y-y1 = slope(x-x1)
remember these derivative rules
power rule
constant rule
product rule (first dlast + last dfirst)
quotient rule (hodihi - hidiho over ho2)
chain rule (don't forget ur peanut bruh)
derivative of sinx
cosx
derivative of cosx
-sinx
derivative of tanx
derivative of cotx
derivative of secx
secxtanx
derivative of cscx
-cscxcotx
derivative of e^x
e^x
derivative of a^x
(lna)(a^x)
derivative of lnx
1/x
derivative of log(a)(x)
1/(lna)(x)
IVT conditions
1) f has to be continuous on closed interval [a,b]
2) f(a) cannot equal f(b)
3) if k is any number between f(a) and f(b)
then the IVT guarantees that there exists at least one number c in [a,b] such that f(c) = k
the IVT guarantees the existence of at least one number c in the closed interval [a,b]
*the ivt often can also be used to locate the zeroes of a function. if f is continuous on [a,b] and f(a) and f(b) differ in sign, the ivt guarantees the existence of at least one zero of f in the closed interval [a,b]
MVT conditions
if f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f'(c) = f(b)-f(a)/ b-a.
CHECK YOUR LIMITS AND MAKE SURE THAT IT IS CONTINUOUS SO GO THROUGH ALL 3 STEPS. YOU MIGHT HAVE TO CHECK LEFT AND RIGHT HAND LIMIT IF PIECEWISE BUT DAS OKAY.
slope of the tan line = slope of the sec line
avg roc = instant. roc
go study the whole curve sketching unit
go study the whole curve sketching unit
derivatives of parametric equations
dy/dx = (dy/dt)/(dx/dt)
so basically find both derivatives and put the y one on top. everytime you wanna find another one, repeat the process and it's probably gonna be a quotient rule but leave dx on the bottom still.
setting dy=0 will find your horizontal tangents.
setting dx=0 will find your vertical tangent or where dy/dx is undefined.
motion along a curve: vectors
<x(t),y(t)> = position vector at any time t
<x'(t),y'(t)> = velocity vector at any time t
<x''(t),y''(t)> = acceleration vector at any time t
your answers will be in <x,y>
how to find the speed of the particle or the magnitude of the velocity vector
square root of dx/dt squared plus dy/dx squared
derivatives of polar equations
x = rcos(x) = f(x)(cosx)
y = rsin(x) = f(x)(sinx)
and then once you find the derivatives of those. bring your parametric boy in and do dy/dx.
you're going to be using your product rules.
riemann sums what to know importantly
don't forget if they're different widths. subtract the two lengths and then use the height depending on the type of sum (left or right).
midpoint. average it out if the x values don't show.
limit definition of a riemann sum
basically find your delta x (b-a/n), then your c(i) (a + i deltax) and then your f(ci) (plug it into your original function).
then find the integral of f(ci) (deltax) <-- you can pull this out of the integral
using your summation formulas
summation formulas
evaluating definite integrals geometrically
you can find the area of a common geometric shape by calculating the areas and adding them together. e.g. semicircles (1/2)(pi r squared) or square or triangles or rectangles, etc.
2 special definite integrals
antiderivatives
always add c
additive integral property
basic integration rules
doing initial conditions and particular solutions
integrate original function. plug in your initial condition and solve for c. and then plug c back into your antiderivative.
fundamental theorem of calculus
mean value theorem for integrals
average value of a function (integral)
integration by u-substitution
decide what you want to consider as your peanut.
then set your own little box u = f(x)
1. choose a sub. on u = g(x) usually the inner part of a comp. function
2. compute du = g'(x)
3. rewrite the integral in terms of u
4. find the antiderivative + c
5. replace u with g(x). back substitute
u-substitution for definite integrals
basically same thing but change your limits of integration by plugging them into the u-function in order to have it in terms of u as well.
ftc
if you change your limits of integration and the one on top ends up being smaller than the one on the bottom, then keep on evaluating you'll be fine
integration by parts
uv − ∫ vdv dx
LIATE in finding u.
let u be the simple one and dv the most complicated portion.
set some boxes to derive u and integrate dv to be able to plug into that upper equation. don't forget + c
tabular method
first sign is always +
use this when power rule multiplied to sinx, cosx, or e^x
partial fractions
this helps you integrate simple functions by that might require u substitution
how to find absolute max / min of a function
1) find derivative of function and find the critical points
2) make a table with endpoints and critical numbers
3) use this table and evaluate the value
4) biggest is your abs. max and smallest is your abs. min
how to find increasing / decreasing on function
1) find derivative
2) find critical numbers
3) after finding critical numbers, put them into a sign chart.
4) test values, plug them into YOUR DERIVATIVE, and if value is positive or neg, that determines whether if it is increasing or decreasing
first derivative test (rel. max/min)
1) find derivative
2) find critical numbers
3) sign charts
4) if it changes from pos to neg, it's a relative max. if it changes from neg to pos, it's a relative min
how to find concavity of a function
1) find derivative
2) find second derivative
3) find critical numbers
4) make a sign chart
5) if it's positive then it's concave up if it's negative then it's concave down
how to find inflection points of a function
1) find derivative
2) find second derivative
3) find critical numbers
4) make a sign chart
5) if it changes concavity then it's an inflection point
second derivative test and relative maximum / minimum
1) find first derivative
2) FIND CRITICAL NUMBERS
3) then find second derivative
4) plug in critical numbers into second derivative
5) if the value ends up being greater than 0, then it is a relative min. if the value ends up being smaller than 0, then it is a relative max.
f'(c) = 0 and one of the conditions on 5 has to be met in order for second derivative test to work. if the second derivative test results in 0 then resort to first derivative test.
how to find total distance
1) zeroes of velocity
2) evaluate the position at end points and critical values (resting points). then subtract the two numbers and take the absolute value of them and add it up for a total distance.
volume of a cube
surface area of a cube
antiderivative of tan
antiderivative of sec
integrating arctan
...
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