Calc part 2

Change of base formula

log base b (a) = log c(a) // log c(a)

Product Property of logs

log b(m \* n) = log b(m) + log b(n)

Quotient Property of logs

log b(m/n) = log b(m) - log b(n)

Power property of logs

log b(m^a) = log b(m) \* log b(a)

Standard position

Angles always start in ____: which is on the positive side of the x axis.

Terminal side

where the angle stops / **terminates**

Quadrantal Angles

Angles that terminate on the x or y axis

Coterminal Angles

angles that share a terminal side

1

180º is equal to ___ pi

Sin (Y/R) // Opposite/Hypotenuse

Cos (X/R) // Adjacent/Hypotenuse

Tan (Y/X) // Opposite/Adjacent

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SOCAHTOA + SYRCXRTYX

Limit

the y value that a function approaches as we approach a particular x value

* *note that the existence of a limit as x→c NEVER depends on how the functions may or may not be defined as c.*

one-sided limits

function f(x) has a limit as x approaches c IFF the right side and left side are equal.

lim x→2- f(x) = 3 (left hand limit)

lim x→2+ f(x) = 4 (right hand limit)

lim x→2 f(x) = DNE

End Behavior

what a function looks like as it approaches infinity or negative infinity (only the highest powers matter for determining this)

given f(x) = 3x^4 - 10x^3 + 2x^2 + 3x + 6

and g(x) = 3x^4,

g is the right hand *__* model iff lim x→(inf) (f(x)/g(x)) = 1

g is the left hand ___ model iff lim x→(-inf) (f(x)/g(x)) = 1

Solving Optimizations

Develop a mathematical model for the problem

Determine the relevant domain of the function

Identify critical points

justify maximums and minimums using the stupid number line thing

State the solution to the problem–make sure to determine whether or not your solution makes sense

Verify graphically if possible

Linarization

if functions are “locally linear” at a point a, you can zoom in really close to a point on the function and the function will look exactly like the tangent line at that point.

* L(x) = f(a) + f’(a)(x-a)
* *Find the Linear approximation of f at a*
* *The estimation for f(4.8) at x = 5 would be L(4.8) = f(5) + f’(5)(4.8-5)*

Differentials

infinitely small changes in x (dx) and y (dy)

It’s just treating dx and dy as variables

y = 3x

d/dx(y) = d/dx(3x)

dy/dx = 3

dy = 3/dx → changes in y are 3 times more large as changes in x

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these can be used to estimate rates of change:

A = (pi)r^2

d/dr(A) = (pi)r^2 \* d/dr

d/dr(A) = 2(pi)r

dA = 2(pi)r \* dr

*then you can plug values in*

Solving Related Rates

Write an equation involving all known and unknown quantities, drawing a picture or diagram may help.

* remember to distinguish constant quantities from variables that will change over time!

Implicitly differentiate both sides of the equation with respect to time (d/dt)

Substitute values for quantities that depend on time (ONLY SAFE TO DO SO AFTER PREVIOUS STEP)

Interpret solution

Integral

Region bound by the x axis and a curve. What is under the curve of a function.

Riemann Sum

an approximation of an integral by using a finite sum

* can estimate the area under a line by using rectangles

lim n→(inf):

n

∑ f(cK)∆x

k = 1

width of each rectangle is ∆x, cK is basically the variable x.

Left hand Rectangular Approximation (LRAM)

estimation technique that makes small rectangles and adds up the areas.

* Each rectangle under the curve f(x) is made of two values a and b (these are dummy variables).
* Height is f(a), width is b - a.
* Underestimates area if the function is increasing. Overestimates if decreasing

right hand rectangular approximation (RRAM)

estimation technique that makes small rectangles and adds up the areas.

* Each rectangle under the curve f(x) is made of two values a and b.
* Height is f(b), width is b - a
* Overestimates if function is increasing, underestimates if decreasing

Midpoint rectangular approximation (MRAM)

estimation technique that makes small rectangles and adds up the areas

* Each rectangle under the curve f(x) is made of two values a and b.
* Height is the average of f(b) and f(a), width is b - a
* Depends if it is overestimate or underestimate

Trapezoidal Approximation Method

estimation technique that makes small trapezoids and adds up the areas.

* Each trapezoid under the curve f(x) is made of two values a and b.
* The area of the trapezoid is found by averaging the two bases and multiplying it by the height. (1/2)(f(b) + f(a)) \* (b - a)
* Usually If the concavity increases, this is an overestimation. Usually if the concavity decreases, this is an underestimation. don’t think you need to know for the exam though.

Sigma notation

lets us express large sums in compact form

n

∑ a(k) = a1 + a2 + a3 + an…

x=1

use this for Riemann sums

Definite Integral

these are numbers, as all continuous functions on \[a,b\] are integrable on \[a,b\].

These are basically the area of a region bounded by a curve and x-axis over some interval. Areas do not have to be positive!

We can estimate the area under the curve by using rectangles, the thinner they are, the more precise they get.

These are the integrals with numbers.

Integral notation

∫(a→b) f(x)dx

a is the lower limit, b is the upper limit

a + K∆x

cK = ???

when converting integral notation to a Riemann Sum or vise versa

(b - a) / n

∆x = ??

when converting a Riemann Sum to a Definite Integral

right, positive

if we integrate and move to the ___ __*,*__ *the region is above the x axis, and the integral value is* __*___.*__

right, negative

if we integrate and move to the ___, the region is below the x axis, making the integral value ___.

left, negative

if we integrate and move to the ___, the region is above the x axis, making the integral value ___.

left, positive

if we integrate and move to the ___, the region is below the x axis making the integral value ___.

Order of integration rule

rule of definite integrals

∫(a→b) f(x)dx == -∫(b→a) f(x)dx

you can switch the upper and lower limits of the integral, you just need to add a negative sign.

Zero rule

rule of definite integrals

∫(a→a) f(x)dx == 0

Constant multiple rule

rule of definite integrals

∫(a→b) k \* f(x)dx = k \* ∫(a→b) f(x)dx

just like every other part of calculus, you can pull out a constant.

Sum/Difference rule

rule of definite integrals

∫(a→b) f(x) ± g(x) dx == ∫(a→b) f(x)dx ± ∫(a→b) g(x)dx

you can break apart parts of an integral and evaluate them separately

Additivity

rule of definite integrals

∫(a→b) f(x)dx + ∫(b→c) f(x)dx == ∫(a→c) f(x)dx

Max-min equality

rule of definite integrals

the minimum of an integral is the lowest point on f ** (b-a), maximum is the highest point on f* \* (b-a)

Domination

given f(x)≥ g(x) on \[a,b\]:

∫(a→b) f(x)dx ≥ ∫(a→b) g(x)dx

if f(x) ≥ 0 on \[a,b\]:

∫(a→b) f(x)dx ≥ 0

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this works if f is above or below, same with g.

Average Value / mean function

Provided f is integrable on \[a,b\] then its average value is:

av(f) = (1/b-a) ∫(a→b) f(x)dx

av(f) being ___.

We can take the average height and find the area of one rectangle instead of finding a lot of rectangle areas and summing them up.

MVT for Definite integrals

If f is continuous on \[a,b\], then at some point c in \[a,b\] f will be equal to the average value of the integral of f:

f(c) = (1/b-a) ∫(a→b) f(x)dx

* *note that the function must reach the average value at some point in time*

First Fundamental Theorem of Calculus

F(x) is f(x)’s antiderivative.

Second Fundamental Theorem of Calculus

F(b) - F(a) = ∫(a→b) f(t)d(t)

Upper limits, First

when plugging in limits of an integral, the ___ will __**ALWAYS**__ be plugged in ___!!!

Indefinite integrals

integrals without a specified upper or lower limit, looking for the antiderivative

## ==DO NOT FORGET TO ADD C!==

Circle Formula

(x-h)^2 + (y-k)^2 = r^2

center being (h,k), radius being r

U-Substitution

taking a part of an integral and setting it equal to u, then finding the antiderivative with respect to u, then plugging back the part of the integral and evaluating.

Slope field

this shows the linearization of a solution to a differential equation at many points.