CP2

What does “Euler’s relation” refer to ?

e^{i theta} = cos theta + i sin theta

What is Euler’s Identity?

e^{i theta} +1 = 0

What is De Moivre’s theorem?

( r (cos theta + i sin theta))ⁿ = rⁿ (cos(n theta)+ i sin(n theta))

Complete these

Where would the 2i sin(n theta) trig identity show up / be helpful in this question?

The denominator will be of the form e^{(pi i / n)} - 1

If we multiply top and bottom by e^{( - pi i / 2n)} the denominator will be of the form e^{pi i / 2n} - e ^{- pi i / 2n}

Which is the same as 2 i sin ( pi / 2n )

What is -1 / i

i

Will i appear in Q ? Why / Why not?

Q is essentially the coefficient of the imaginary part

therefore DO NOT include i

Is this the general form for the n^th roots of unity?

No

It is only 1 root

The power should be 2 k pi i / n

What is the sum of the n roots of unity? Why?

0

Adding complex number is like adding vectors.

The vectors are equally spaced so cancel each other out.

When we sum them, the imaginary parts all cancel (conjugate pairs) and the real parts will also lead us to 0

If we have one root of a complex number, e.g. z₁ , how do we find the remaining roots of zⁿ ?

The n^th roots of a complex number form the vertices of a regular n-gon centered where?

The origin

When is the Maclaurin series expansion invalid?

If at least one of the derivatives at x=0 is undefined.

If the series does not converge to the true value of the function

What can you use to signify an alternating sequence?

(-1)ⁿ

True or False? You must use radians for the trig Maclaurin series expansions

True!

In terms of Improper integrals, what does convergent refer to and mean in context?

The limit of the integral,

The improper integral exists

An improper integral exists if the limit converges, and does not exist if the limit diverges.

In terms of the function, what determines if the area converges/ diverges?

The rate at which the function grows near the boundary

How does the mean value of the function change if we:

add some constant to the function

Multiply the function by some constant

Reflect the function in the x-axis

What do the inverse trig graphs look like and why would this be relevant in the methods in calculus topic?

The domain and range is important for integrating and differentiating the inverse trig functions

What did you forget on this question?

You need to mention that arc cos x has a range of 0 to pi

and sin is positive in this interval

hence we have the positive square root

When we let x = a sin u in this question, what is the implication for the range of u?

Why is this important?

-pi/2 < u < pi/2

When we get to sqr cos² u we know to take cos u to be positive because cos is positive over the given domain

Give THREE examples of odd functions and TWO examples of even functions

EVEN: cos, cosh

ODD: sin, sinh, tan

What is the name of the rule used to find hyperbolic identities from trig identities?

What is this rule?

Osborn’s rule

Change cos to cosh and sin to sinh,

If there is a explicit or implicit product of two sinh then change the sign

What form does the differential equation have to be in before you find the integrating factor?

What does a “linear first order differential” look like?

What does a “Linear homogeneous second-order differential equation” look like?

How does a non-homogeneous differ?

With a non-homogenous the RHS is not 0

How do you solve a Linear homogeneous second-order differential equation ?

Rearrange the differential equation so that you can

Solve auxiliary equation

which determines the general solution

What is the general solution if the roots of the auxiliary equation are imaginary?

What are the particular integrals if the general solution is of these forms