General

Question 1

State the equation for the Taylor series

Answer 1

f(x) = ∑^∞_{n=0} f^{(n)}(a)/n! (x-a)^n

Here, f^{(n)}(a) denotes the n-th derivative of f(x) evaluated at x=a (often x=0 is used), and n! represents the factorial of n.

Question 2

What is a taylor series used for?

Answer 2

Evaluating the value of a whole function at each point if functional values and derivatives are known at a single point. Used for approximating functions.

Question 3

What is the Taylor series expansion of f(x) = e^x ?

Answer 3

e^x = 1 + x + x^2/2! + x^3/3! + …

Question 4

What is the Taylor series expansion of f(x) = sin( x ) ?

Answer 4

sin( x ) = x - x^3/3! + x^5/5! - …

Question 5

What is the Taylor series expansion of f(x) = cos( x ) ?

Answer 5

cos( x ) = 1 - x^2/2! + x^4/4! - …

Question 6

What is the Taylor series expansion of f(x) = 1/{1-x} ?

Answer 6

1/{1-x} = 1 + x + x^2 + x^3 + … for |x|< 1

Question 7

State the equation for the Binomial Theorem of (a+b)^n?

Answer 7

(a + b)^n = ∑^n_{k=0} (n on top of k) a^{n-k}b^{k}, where (n on top of k) = n! / {k! (n - k)!}

Question 8

Expand sin(A + B)

Answer 8

sinAcosB + cosAsinB

Question 9

Expand cos (A + B)

Answer 9

cosAcosB - sinAsinB

Question 10

What is coshiz equal to?

Answer 10

cosz

Question 11

What is is sinhiz equal to?

Answer 11

isinz

Question 12

State the quadratic formula to find the roots of ax^2 + bx + c = 0.

Answer 12

x= ( - b ± √ { b^2 - 4ac } ) / 2a.

b^2 - 4ac > 0: 2 real and distinct roots.
b^2 - 4ac = 0: 1 real and repeating root.
b^2 - 4ac < 0: 2 complex roots.

Question 13

If ∫_0^∞ f(x)dx is even, i.e. f(x) = f( - x), what can happen to the limits of integration?

Answer 13

They flip and infinity becomes negative, such that ∫0^∞ f(x)dx = ∫- ∞^0 f(x)dx