ap calculus bc semester 1 review

area of trapezoid

(1/2)(b1 + b2)(h)

area of triangle

(1/2)(bh)

derivative of a^x

(lna)(a^x)

derivative of cscx

-cscxcotx

derivative of cosx

-sinx

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

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

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]

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)

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

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 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

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

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)

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

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.

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

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

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.

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.

derivative of log(a)(x)

1/(lna)(x)

derivative of lnx

1/x

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>

speed is the ______________ of velocity

ABSOLUTE VALUE

antiderivatives

always add c

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

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

when velocity changes signs, the object

changes direction

derivative of sinx

cosx

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

differentiability and continuity

differentiability implies continuity

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.

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.

derivative of e^x

e^x

tabular method

first sign is always + use this when power rule multiplied to sinx, cosx, or e^x

go study the whole curve sketching unit

go study the whole curve sketching unit

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

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

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.

if velocity is negative but acceleration is negative

it's moving left and gaining speed

if velocity is negative but acceleration is positive

it's moving left and slowing down

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

three special limits 3)

limit as x approaches 0 = (1+x) raised to (1/x) = e

area of rectangle

lw

volume of rec. solid

lwh

l'hopital's rule

only if indeterminate

if concavity is downwards

overestimate

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)

formal definition of a derivative

replace with delta x

derivative of secx

secxtanx

when acceleration is 0

speed is CONSTANT

when velocity and acceleration have different signs

speed is DECREASING

when velocity and acceleration have same signs

speed is INCREASING

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

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)

when velocity is at 0

the object is at rest

partial fractions

this helps you integrate simple functions by that might require u substitution

if concavity is upwards

underestimate

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

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.

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.