Calculus

Question 1

Definition of a derivative

Answer 1

Question 2

Top Derivatives and Integrals

Answer 2

Question 3

Product Rule and Quotient Rule

Answer 3

Question 4

Inverse functions derivative rule

Answer 4

Inverse functions are symmetrical about the line y = x

Question 5

Chain Rule Differentiation

Answer 5

Question 6

Implicit vs. Explicit Differentiation

Answer 6

explicit: y = 2x
implicit: y = 2x + y² (def of y involves y)

To solve implicit derivatives, put in (y’) where appropriate and solve for y’

example:

Question 7

What are critical values of a derivative, and what is their importance?

Answer 7

Where the derivative equals zero. They can be relative/global maximums/minimums.

Use the first derivative test to determine if they are max or min.

Use the second derivative test to determine if they are concave up or concave down (which tells you max or min)

The second derivative test also tells you if a point is an inflection point or not (the 2nd derivative will equal zero).

Inflection points are where functions transition between concavity.

These tests can be applied to any order of a derivative

Question 8

At what points are derivatives undefined in a function?

Answer 8

A cusp, a vertical tangent, and discontinuity

Question 9

Mean Value Theorem

Answer 9

If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) such that: (view image)

Explanation: a secant line that is drawn from point a to b has a slope, m

There has to be some point, c, whose tangent line slope is also equal to m

Question 10

Visual depiction of the mean value theorem

Answer 10

Question 11

How to optimize dimensions (and other stuff) through differentiation

Answer 11

1. Express the problem as a formula
2. Determine the domain
3. Find the critical numbers of the derivative
4. Test the critical numbers and the endpoints of the domain for the maximum value of the original formula

Question 12

What are related rates?

Answer 12

Example: filling up a pool with a water hose

The rate of depth in the pool is related to the rate of water out of the hose

The key is to set up an equaiton, then differentiate both sides, then solve implicitly to get your desired rate

Question 13

How to use linear approximation at a point

Answer 13

Find the slope (through the derivative)

plug into:

y - y₁ = m (x - x₁)

Gives you a line to process local approximations

m is the same as f ^’ (x)

Question 14

What is the meaning of “marginal” in these cases: marginal cost, marginal revenue, marginal profit

Answer 14

The increase or decrease per 1 unit over (x + 1)

Linear approximation works great for this

Question 15

Left, Right, and Midpoint area calculation formulas (All are considered Riemann sums)

Answer 15

n is the number of rectangles, (b-a)/n is the width of each triangle, function values are the height of each rectangle

Question 16

Summation Notation

Answer 16

Question 17

Definition of an integral

Answer 17

Where delta x is the withd of the ith rectangle and x_i is the x-coordinate of the point where the ith rectangle touches f(x)

Question 18

Another way to calculate the area under a curve that isn’t Reimann sums is the trapezoid rule

The trapezoid rule is basically the average of the left and right reimann sums

Answer 18

Where n is the number of trapexoids, x₀ equals a, and x₁ through x_n are the equally-spaces x-coordinates of the right edges of trapezoids 1 through n

Question 19

Simpson’s rule for calculating the area under a curve

Answer 19

The most accurate against the riemann sums and the trapezoid sums.

Bassically, its an average of the midpoint sum counted twice, and the trapezoid sum.

n is twice the number of “trapezoids” and x₀ ans x_n are the n + 1 evenly spaced points from a to b

Each “trapezoid” spans two intervals instead of one; in other words, “trapezoid” number 1 goes from x₀ to x₂ “trapezoid” 2 goes from x₂ to x_4. Because of this, the total span must always be divided into an even number of intervals

When calculating, multiply the outside term of the brackets by each inside the brackets

Question 20

Fundamental Theorem of Calculus

Answer 20

Question 21

Substitution Method for integration

Answer 21

It’s so natural in your mind, it seems pointless when you read the explanation

Question 22

Integration by parts

Answer 22

Question 23

The Mean Value Theorem for Integrals

Answer 23

Useful for finding the average value of the funciton over a particular interval

f(c) is the average value of f(x) over the interval [a, b]

Question 24

How to find the area between two curves

Answer 24

Find the two intersection points.

Evaluate the top function’s integral over that interval.

Evaluate the bottom function’s integral over that interval.

Find the difference. That difference equals the area between the curves.

Question 25

How to find the volume of weird shapes using integration

Answer 25

1. Find the area of a normal “slice”
2. Tack on the “dx” to the equation for the area
3. Integrate that whole expression over the desired interval = volume

This method can be used to solve the difference in volumes of two shapes as well

Good test: Find the volume of the solid bounded by x = 2, x = 3, and y = e^x. Rotated around the y-axis. Answer: 206.

Explanation: Each “slice” is 2*pi*x * e^x (think wrapper around can). Add the “slices” from radii 2 to 3

Question 26

How to calculate the arc length of a curve with an integral

Answer 26

Question 27

How to find the surface area of a shape created through revolutions

Answer 27

Take the tiny cylinder wrapper area, integrate it over the desired axis

Notice the expression for arclength is included

Question 28

How to simplify constants within an integral

Answer 28

Remember you can take the constant outside of the integral

Question 29

L’Hôpital’s Rule

Answer 29

Great shortcut for doing limit problems

Let f and g be differentiable functions. If the limit of (f (x) / g (x)) must be 0/0 or infinity / infinity for this to apply. Call this “indeterminate.” The limit will equal this:

Question 30

For limits, what are some unacceptable forms

Answer 30

+ / - infinity * 0

infinity - infinity

1 ^{+ / - infinity}

0⁰

infinity ⁰

For the last three cases, you can use the natural log trick:

Set the limit to “y” , then take the natural log of both sides until you can get it to an acceptable condition of L’Hospital’s rule, then solve for y.

Question 31

What makes a definite integral improper

Answer 31

If they go infinitely up, down, right, or left and can’t be calculated

Question 32

Special notes about determining area for improper integrals that have vertical asymptotes

Answer 32

Some you can, some just go to infinite… Here’s how you do it

Overall, you need to durn the definite integral into a limit

The first improper integral diverges, the second integral converges (there is an actual finite number as the answer)

These examples are ones where the vertical asymptotes are at the limits of integration. If the vertical asymptote is in between the limits of integration, you need to split it up into two separate limits about the undefined point. Be sure one limit is evaluated from the left, and the other limit is evaluated from the right. If any component of this divided integral diverges, the original whole version diverges.

Question 33

Another way for improper integrals to converge or diverge

Answer 33

Besides having vertical asymptotes, improper integrals can diverge or converge from infinty set as one of the limits of integration

(The example converges)

If both of the limits of integration are negative infinity and positive infinity, then split it up into two integrals at 0 and solve. Doesn’t have to be 0…

Question 34

Difference between a sequence and a series

Answer 34

Series is sum of the terms

Question 35

What does it mean for a sequence to converge

Answer 35

The sequence approaches a finite term as n approaches infinity.

If it doesn’t, it’s considered divergent. This happens when the sequence increases forever, oscillates, or has no pattern at all

Question 36

Using L’Hopital’s rule with sequences

Answer 36

Use the rule twice over (find the 2nd derivative of the numerator and denominator). Evaluate the limit at infinity, and it will show you where the sequence will converge, if at all.

Question 37

What are partial sums of an infinite series

Answer 37

A lot of times you’re asked to find the limit of the partial sums (as their own sequence)

Question 38

What’s special about a sequence of partial sums

Answer 38

Every series and its related sequence of partial sums are either both convergent or both divergent, moreover, if they’re both convergent, it’s to the same number

Question 39

Divergence test example

Answer 39

If the individual terms of a series do not converge to zero, then the series must diverge

Question 40

P-series rule

Answer 40

Question 41

Telescoping series rule

Answer 41

The sum ends up being 1 + 1 / (n + 1)

Which is either some number between 1 and 0, or 1

Question 42

Limit comparison test

Answer 42

Question 43

Direct comparison test

Answer 43

This tells you nothing if the series is bigger than a known convergent series or smaller than a known divergent series

Question 44

Integral Comparison test

Answer 44

Basically, if you can integrate the series expression, do so, and see if the integral converges or diverges

Question 45

Ratio test

Answer 45

The ratio of a geometric series has to be less than 1 to converge

Question 46

Root test of a series

Answer 46

Question 47

How to integrate difficult polynomial expressions

Answer 47

Do partial fraction expansion, and integrate each individual part

Question 48

Alternating series: absolute vs. conditional convergence

Answer 48

Alternating series change signs every term. If left alone, and if the series converges, the series is said to be conditionally convergent. If all the terms were made into the same sign, and the series was still convergent, then the series would be called absolutely convergent

Question 49

The alternating series test

Answer 49

Question 50

The nth term test for series divergence

Answer 50

Question 51

Keeping all the series tests straight

Answer 51

10 tests mentioned total

Geometric series, p-series, and telescoping series. A geometric series convergesif r is between 0 and 1. A p-series converges if p > 1. A telescoping series converges if the second “half term” converges to a finite number.

Comparison tests: direct comparison, limit comparison, and integral comparison tests. All three compare a new series to a known benchmark. If the benchmark converges, so does the series you’re investigating; if the benchmark diverges, so does your new series.

Two “R” tests. Ratio and root. No benchmark comparison, Both involve taking a limit, and the results are analyzed. If the limit is less than 1, the series converges; if the limit is greater than 1, the series diverges, and if the limit equals 1, the test is inconclusive.

The nth term test of divergence and the alternating series test.

Question 52

Taylor series wordy explanation

Answer 52

The taylor series is a specific form of the power series. It can be thought of a polynomial with an infinite number of terms. The interval of convergence of a power series is the interval of x values that will converge to a number.
The Maclaurin series is a simplified version of the Taylor series.

Question 53

Format of a power series

Answer 53

Question 54

Power series general form

Answer 54

Centered around “a”

Question 55

Properties of the interval of convergence

Answer 55

Every power series converges for some value of x. The interval is never an empty set

Use the ratio test to determine if so (taking limits)

Question 57

Maclaurin Series

Answer 57

1. Find the first few derivatives of the function until you recognize a pattern
2. Substitute 0 for x into each of these derivatives
3. Plug these values, term by term, into the formula for the Maclaurin series.
4. If possible, express the series in sigma notation

Question 58

Taylor Series

Answer 58