Cumulative Calculus

What is equivalent to sin(3x)^2

(1- cos (6x))/2

FInd the derivative of ln(lnx)

(1/x)/ln(x)= 1/(xlnx)

Derivative of b^x

(lnb)b^x * x'

Which of the following is the solution to dy/dx=e^(y+x) with y(0)= -ln(4)

-ln(-e^x+5). REMEMB ER: when taking ln of both ides, take ln of (ab), not ln(a)+ln(b)

Integral of -tan(X)

-tan = -sin/cos, which is the same as the derivative over the function, which is the same as ln(cos(x))

Link: https://www.trussvillecityschools.com/Teachers/Ryan.James/AP%20Calculus%20BC/AP%20Exam%20Review%20Materials/AP%20Cal%20BC%20S2%20Most%20Missed%20Midterm%20Questions.pdf

...

How to approach limit problems

1.) direct substitution 2.) factoring and reducing 3.) rationalizing 4.) squeeze theorem for sin/cos **5.) Using the properties sinx/x=1 and 1-cosx/x=0**

Intermediate Value Theorem explanation

1.) function must be continuous within an interval 2.) the two y values at the tips of the interval can't be the same number 3.) there must be an x value that denotes a y value between the values of the endpoints. 4.) Alternately, if two y values have opposite signs within a closed interval, there must be at least one zero within that interval

Area under a polar curve

1/2 integral from a to b of (r)^2

menu of cos(6x)

1/6sin(x)

What is the derivative of the inverse of f(x)?

1/f'((g(x)) in which g(x) is the inverse. So it's one over the derivative of the original function with the inverse where x should be.

integral of 4^u

4^u/ln(4) in which u is a function

What is e^2ln(1+x)?

= e^ln(1+x)^2 = (1+x)^2

What is the definition of continuity at a point

A function is continuous at a point if the limit as x approaches c equals the y value at c.

What is a monotonic function?

A function that is only increasing or only decreasing on an interval. The derivative will be positive or negative.

The power series sigma An(x-3)^n converges at x=5. Which of the following must be true? A. diverges at x=0 B.diverges at x=1 C. converges at x=2 D. converges at x=6

A series always converges at its center, which in this case is 3. But since this function also converges at 5, that means its radius of convergence extends to at least two integers in each direction. However, because there is no way of checking the bounds, it is unknown whether the series diverges at 1 or not. It is also unknown whether the radius of convergence is greater than 2. All you know that MUST be true is that it must converge at 2.

Are all critical points extrema? Are all extrema critical points?

All extrema are critical points or endpoints, but some critical points are not actually extrema. Like f(x)=x^3 has a derivative equal to zero at zero, but x is not a maximum or a minimum. This means that if you actually want to find the max and min of an interval, you have to compare the critical points to the end points to see which is the actual lowest or highest. Remember to get rid of critical points that are not within the interval.

The FTC says that there is one condition for integration

As long as the function is continuous, it's integratable

When is a function continuous but not differentiable?

At a cusp. A function will not be differentiable at a point where gradient of the function approaching the point from one side has a different value as you approach the point from the other side - so you get a pointy bit! Others functions that are not differentiable are sqrtx at x=0 and x^(1/3) at x=0. That's because the secant lines approach a vertical line at those points.

Where does a critical point exist?

At a point that is nondifferentiable or at a point where the derivative equals zero.

f'(x)= x^3-5x^2+e^x. When is f concave down?

Because you're given the derivative and asked to find the second derivative, use the values when the SLOPE of f' is negative. Here, from (0.116, 2.062)

What is approximation by linearization?

Finding a simpler equation for which you can plug in numbers and get y values similar to those of the original equation. You find the linearization at a POINT, so that all the points plugged in around that point will be close to the original . L(x)= f'(a)(X-a)+f(a) Linearization equals the derivative at that point times x minus that point, plus the original equation at that point. Same thing as linear approximation. Derivative times difference in x, only to find the VALUE rather than the difference in values, you add the original f(a).

Calculate the derivative at 1 of the inverse of f(x)= x+e^x

First of all, f'(x)= 1+e^x. Now we don't know what the the inverse of that function would be, but we know that 1=x+e^x because you put it as a y value, since the domain and range of the inverse are flipped. THis gives us x=0, which you now plug into 1/(1+e^x) to get 1/2.

Calculate the derivative of the inverse of x^4+10

First of all, the inverse of the function is (x-10)^(1/4), which means the domain must be greater than zero. The derivative of the inverse is 1/4x^3, in which x= (x-10)^(1/4). Therefore, it is 1-4[(x-10)^(1/4)]^3

Use Rolle's theorem to prove that an equation has only one real root. (aka it crosses the x axis exactly once)

First prove there there is, in fact, a root by finding a negative and a positive y value and saying that there must b a root between them by intermediate value theorem. Then, prove that there aren't other roots by trying to find a place where the derivative equals zero. Since there aren't any places where this is true, there are no places where the y values are the same, including no other place where the y value is zero ( a root).

Find the derivative of y=x^sinx

First take the ln of both sides ln(y)= ln(x^sinx) Now rearrange ln using properties ln(y)=sinx*ln(x) Take the derivative of both sides with respect to x using multiplication rule (1/y)y'=sinx(1/x)+ln(x)(cosx) Now arrange everything so y' is one one side and substitute y for the original equation. y'= [sinx(1/x)+ln(x)(cosx)]x^sinx

Difference between integrating for displacement and integrating for total distance

For displacement, just use your normal bounds and things will cancel themselves out. For total distance, find where there are roots in the function and find the absolute value between each set of roots. Add the abs values together for total distance.

Mean Value Theorem

Function must be continuous and differentiable on closed interval. Theorem says that there must be an x value at which the slope of the tangent line is equal to the slope of the secant line of the endpoints

Which of the following series converge? I. abs(sin n)/n^2 II. e^-n III.n+2/(n^2+n)

I and II only

Easiest way to test if a series diverges.

If the limit as n approaches infinity is not zero, it diverges for sure. If it is zero it may either converge or diverge.

review: derivative of inverses

If you find the slope of the tangent line of the function, the slop of the tangent of the inverse is 1/original deriv. NOT negative.

Find the derivative of e^(x-y)=2x^2-y^2

Implicit differentiation. Find the dy/dx of both sides. e^(x-y) * (1-dy/dx)=4x-2y(dy/dx) e^(x-y) * 1-e^(x-y)(dy/dx)=4x-2y(dy/dx) -e^(x-y)(dy/dx)+2y(dy/dx)=-e^(x-y)+4x (dy/dx)(-e^(x-y)+2y)=-e^(x-y)+4x dy/dx= (-e^(x-y)+4x)/(-e^(x-y)+2y)

What determines speed increase or decrease?

Increase is when velocity and acceleration are both pos or both neg. A mix of the two leads to speed decreasing

"find the total distance traveled" when given the velocity

Integrate the ABSOLUTE VALUE of the velocity over the interval. Do this by finding the zeroes of the velocity. Don't forget to add the initial distance traveled if applicable.

What happens if a limit approaches infinity?

It does not exist

If you get zero in the denominator for a limit...

Keep working, there's an answer. You may have to plug in to find that it's infinity or negative infinity.

Integral of a fraction

Life changing formula or integration or partial fractions.

Integral of a semicircle

Make sure that the function that you're plugging in as "r" and squaring is actually that of the RADIUS. Sometimes the function for the diameter will be given. So you will have to divide by two inside the squaring parenthesis to get the radius, and outside the integral to get half the area of the full circle.

What is linear approximation?

Method used to estimate the change in y values by using the derivative. dy/dx= fx becomes dy=f'(x)(dx). Dx represents the change in x, and dy is close to the value of the change in y when dx is small. Think of it as new y = old y + change in y, in which you get the change by y by multiplying dy/dx times the change in x to cross out dx and get dy.

What are the two conditions of the interval of the extreme value theorem?

Must be continuous and must be on a closed interval. Why? Because an open interval allows it to go to infinity, and something like a jump discontinuity at the peak would meant that there isn't, in fact, a maximum. The theorem states that if the two above are true, c will take on BOTH a minimum and a maximum value.

Find the derivative of 1/lnx

Must use quotient rule. (lnx)(0)-(1)(1/x)/lnx^2

Find the maximum population of dy/dx = P(2- (P/5000))

Not in logarithmic form because a 1 needs to be at the front Divide both by two: dy/dx = 2P ( 1 - (P/10,000)) and max population is 10, 000

If a function is undefined at a certain point, what does that say about the limit?

Nothing, it can still approach a number provided that the limit is the same from the left and right. Thus, a jump discontinuity does not affect the limit.

Which of the following is he Maclaurin series for 1/(1-x)^2?

Remember that you can take the derivative of ever term, but you cant square every term. So since the derivative of 1/(1-x) is 1/(1-x)^2, take the derivative of x+x^2+x^3 to get 1+2x+3x^2

Disk method trick

Remember to square each equation you are using separately before subtracting them. You only subtract before squaring to find regular area without volume.

Find the derivative of sin3x

Remember to use chain rule--> derivative of outer function with inner function inside * derivative of inner function. 3cos3x

Find the magnitude of the velocity vector

Same as the speed of a vector: pythagorean theorem using the DERIVATIVE of the position functions for x and y. (formula of velocity and speed are the same because the speed is the magnitude of the velocity)

Limit operations rules

Sum and quotient: limx-->c(f(x)+g(x))= limx-->cf(x)+limx-->xg(x) Product and quotient law: limx-->c(f(x) ** g(x))= limx-->cf(x) ** limx-->xg(x) Power law: limx-->c(f(x)^2)= (limx-->cf(x))^2

Discuss the continuity of this piece wise function: F(x)= x when x is less than one f(x)=3 when x is between 1 and 3 f(x)=x when x is greater than three

The limit at one from the left is one, but the limit at one from the right is three, so there is a jump discontinuity at one. However, the limit from the right at 3 is 3, and so is the limit from the left, so the function is continuous at x equals 3.

A tank has 50 liters of oil at t=4/ R(t) is the rate a which oil is pumped. Using a right Riemann sum with three intervals, what is the number of oil at t=15 hours>

The mistake: didn't add the 50 liters from the beginning to the Riemann sum.

What does d^2y/dx^2 mean?

The second derivative using implicit differentiation

What do limits at infinity tell you about the function

They give you horizontal and vertical asymptotes. Plug in a really large or small number into the leading terms to find out. If the leading coefficients are the same, it'll give you a number that it's approaching.

The fuel consumption of a car, in miles per gallon, is modeled by F(S) = 6e^(s/20 - s^2/2400) where s is the speed of the car in mph. If the car is traveling at 50 mph and its speed it changing at the rate of 20 mph^2, what is the rate at which its fuel consumption is changing? (Answers are in mpg per hour)

To get mpg per hour you have to differentiate the mpg function (F(S)) in terms of hours. You do this by implicit differentiation and using ds/dt which is given to you as 20 (derivative of speed in relation to time) to do it.

If a and b are positive constants, the limit as x approaches infinity of ln(bx+1)/ln(ax^2+3) is

Use L'hopital's rule to get b/(bx+1)times ax^2+3/ax Then simplify the X^@ to get 1/2

Find the derivative of logx

Use change of base formula. lnx/ln(10)= (1/ln10) * ln(x)= (1/ln10) * (1/x)= 1/(ln10 * x)

What is the second derivative test?

Using the second derivative to tell if a function is concave up or concave down. When a curve is concave down (max), the second derivative is negative because the slope of the tangent line started out as positive, hit zero, and is now decreasing. When a curve is concave up (min), the second derivative is positive because the slope of the tangent line started as negative, hit zero, and is now increasing.

What is the first derivative test?

When you set the derivative to zero to find critical points, the make a number line from left to right to see if it goes from positive to negative or negative to positive. If it goes from neg to pos, you have a local minimum. Pos to neg is a local maximum.

Find the cube root of 28 using linear approximation

You know the cube root if 27 is 3. That means your original function is y=(x)^(1/3) dy=1/(3 ** (x^(-2/3)) ** dx The change in x is 1, and the the original x is 27. Substitue these. Your answer will be around .037, but that is not the function value at 27. To find that out, add .037 to 3, and you will have an approximation of y at 28.

If f'(x) > 0 for all real numbers and the integral of f(x) from 4 to 7 equals zero, which could be a table of values for f?

You need to find a table that shows a triangle across the origin in which the left and right sides cancel each other out. So the table of values MUST be increasing, and MUST cross 0.

Prerequisites for Rolle's Theorem and Mean Value theorem

You're looking for a place where the derivative is something specific, so it must be continuous and differentiable on the closed interval.

Prerequisite for alternating series test

a^n must be positive (greater than zero) and a^n+1 < a^n (decreasing)

Some nondifferentiable functions

absolute value of x, jump discontinuity, point discontinuity, infinite discontinuity, y= x^(1/3), y=sqrtx at x=0,

Write the third-degree Taylor polynomial

actually write out several terms

Find the interval of convergence

check interval!!

What kind of functions always have a maximum AND a minimum

continuous (no holed) and bounded (no infinity) functions

If x= t^2-1 and y = ln(t), what is dy^2/d^x?

derivative of top is -1/t and of bottom is 2t to make -1/2t^2. Second derivative is -1/2t^4

Logarithmic growth:

dt/dt = ky(A- y) and y = A/(1+Ce^-Akt) OR dt/dt = ky(1- y/A) and y = A/(1+Ce^-kt) mnemonics: the one about A being at the forefront, and the one about the subtraction only coming into the exponent. Think of logarithmic growth as the next step to the exponential dy/dt = ky

speed of a particle using vectors

dy/dt and dx/dt are the velocities of the horizontal and vertical motions. To find the speed of the vector itself, use pythagorean theorem. sqrt( (dy/dt)^2 + (dx/dt)^2). Make sure to use the DERIVATIVE

For the integral of sin x from 0 to 2pi, you must remember to..

evaluate -cos at 0 because it's an actual vale to subtract from.

What is the derivative of f(g(x))?

f'(g(x)) * g'(x)

Derivative of ln(f(x))

f'(x)/f(x)

3 conditions for continuity

f(c) is defined, lim as x approaches c exists, the two are equal to each other

derivative of the inverse

if g(x)= inverse f(x), then the derivative if g(b) is 1/ f(g(b)) (literally one over the derivative of thr inverse)

lim of sinx/x as x approaches infinity. what about as x approaches zero?

infinity: use squeeze theorem to find that both 1/x and -1/x are approaching zero. since it is at infinity, it would be a horizontal asymptote. zero: use lhopitals rule to get 1. this would be testing at a soecific point, but it is not A vertical asymptote

length of the arc of a vector

integrate the pythagorean theorem using the DERIVATIVE of the position functions for x and y. How to remember which one is integrated: speed is the integral of distance. So the integral of this is arc length, and the "derivative" (the normal function) is speed.

Know what log and exponential functions look like

look it up. But you know when x equals zero e^x =1, and ln crosss the x axis at 1

If your integration bound are from x to 3...

make it easier to solve by pulling out a negative and making the bounds from 3 to x.

derivative of the integral from -2 to x^2 of sin(x)

sin(x^2)**2x - sin(-2)**(0) = 2xsinx^2

Derivative of the inverse of trig functions (arc-) sin cos tan sec

sin--> 1/sqrt(1-x^2) cos--> -1/sqrt(1-x^2) tan--> 1/(1+x^2) sec--> 1/abs(x)sqrt(x^2-1)

derivative of trig funtions sin cos tan sec cot csc

sin= cos cos= sin tan= sec^2 sec= sectan cot= csc^2 csc= cotcsc

if f(3)=5 and f'(3)=7, what is the line tangent to the inverse of f (f^-1)

switch the points to get 5,3. the slope is 1/7. y=3+(1/7)(x-5)

Definition of the limit

the limit as x approaches a --> (f(x)-f(a))/(x-a) OR limit as h approaches zero --> (f(x+h)-f(x))/h where h is a really small quantity

Mean Value Theorem for Integrals

the mean value of the function from a to b is the integral from a to b divided by (b - a). When doing problems like this, mean sure that the "mean value" you find is actually between b and a

What is the slope tangent to the curve r= 1+ sin(t) at x=(t)

the slope will be dy/dx, so need to find the derivative of x and the derivative of y.However, you must first determine x and y by using x=rcos and y=rsin. Then find that value at zero.

"what is the averge rate of change..."

use slope formula

Integral of ln(x)

x ln(x) - x + C.

Functions that are always continuous

y= a constant y= x^b(when the exponent is a whole number, aka polynomial function)

Exponential growth

y= y0 e^kt OR y= y0 e^-kt (for decay) in which y0 is the original, k is a constant, and t is the amount of time passed. dy/dt = ky

don't forget critical points where x DNE

yup

Find the derivative of ln(sqrt(sinx))

{[0.5(sinx)^-0.5]*cosx}/sqrt(sinx)