Midterm Review

Question 1

Define Function

Answer 1

A rule that assigns each x in a set D a unique element f(x) in a set R.

Question 2

Rules for Inequalities

Answer 2

1. If a < b, then a+c < b+d
2. If a < b, c < d, then a+c < b+d
3. If a < b, then ca < cb if c > 0
4. If a < b, then ca > cb if c < 0
   4a) If 0 < a < b, then 1/a > 1/b
   4b) If a < b < 0, then 1/a > 1/b

Question 3

How to solve Inequalities with double solution? (x-a)(x+b)>0

Answer 3

Use sign chart, ie test each interval (x < b, b < x < a, x > a) to see which is

Question 4

Properties of Absolute Value

Answer 4

√(x^2) = |x|

1. |ab|=|a||b|
2. |a/b|=|a|/|b| (b != 0)
3. |x| = a, or x = +/-a
4. |x| < a, where -a < x < a
5. |x| > a, where x > a or x < -a

Question 5

Define Even Function

Answer 5

Where f(-x) = f(x) for every x in the domain. Even functions are symmetric wrt y-axis.

Question 6

Define Odd Function

Answer 6

Where f(-x) = -f(x) for all x in the domain. Odd functions are symmetric wrt the origin.

Question 7

1:1 Functions

Answer 7

If a function never maps to the same value twice. Horizontal line test.

Question 8

Inverse of Functions

Answer 8

Inverse DNE for non 1:1 functions. To find inverse, sub y for f(x) and solve function for x, then switch x with y. Draw the graph of inverse by reflecting function wrt the line y=x.
Domain f = Range f^-1, Range f = Domain f^-1

Question 9

Periodic Functions

Answer 9

f(x) is periodic if f(t+nT)=f(t) for all integer n.
Frequency: f= 1/T Hz
Angular Frequency: w=2πf

Question 10

Hyperbolic Functions

Answer 10

cosh(x) = (e^x + e^-x) / 2, sinh(x) = (e^x - e^-x) / 2

Question 11

Logarithmic Functions

Answer 11

If a^c = b, then log[a]b = c
If f(x) = a^x, then f^-1(x) = log[a]x
If f(x) = e^x, then f^-1(x) = lnx

Question 12

Rules of Logarithms

Answer 12

1. loga = log[a]x + log[a]y
2. loga = log[a]x - log[a]y
3. log[a]x^v = vlog[a]x

Question 13

Polynomials

Answer 13

Degree = largest power term.

Rational function is proper if degree top < degree bot

Question 14

Partial Fraction Expansion

Answer 14

Proper rational function can be written as sum of partial fractions with form A/(ax+b)^2 and (Cx+D)/(ax^2+bx+c)^2 ~ irreducible factor.
For any factor repeated n times, you need n terms of the form given distinguished by exponents 1-n.
Solve by multiplying out the denominators and setting x to be a value which eliminates one constant.

Question 15

How to go from Improper to Proper?

Answer 15

long division until you get poly + proper

Question 16

Distance between points

Answer 16

√((x2-x1)^2+(y2-y1)^2) or √(Δx^2+Δy^2)

Question 17

Equation of a Circle

Answer 17

x^2 + y^2 = r^2 or (x-h)^2 + (y-k)^2 = r^2 whose center is at point (h,k) and radius r.

Question 18

Arcsinx

Answer 18

Domain= [-1, 1], Range= [-π/2, π/2]

Question 19

Arccosx

Answer 19

Domain= [-1, 1], Range= [0, π]

Question 20

Arctanx

Answer 20

Domain= (-∞, ∞), Range= [-π/2, π/2]

Question 21

How to add two sin/cos waves of same frequency?

Answer 21

If asin(wt) + bcos(wt):
Asin(wt + θ)
A = √(a^2+b^2)
tanθ = b/a

Question 22

Sequences

Answer 22

A sequence has limit L if for every +ive real # ε there exists an integer N such that for all n > N we have
|a[n] - L| < ε

Question 23

Squeeze Theorem

Answer 23

If f(x) ≤ g(x) ≤ h(x) for x≥N and lim[x→∞]f(x) = y, lim[x→∞]h(x) = y, then lim[x→∞]g(x) = y

Question 24

Important Limit and Indeterminate Forms

Answer 24

lim[x→∞] sinx / x = 1

∞-∞, 0/0, ∞/∞, 0x∞

Question 25

Continuity

Answer 25

f(x) is continuous at a IFF lim[x→a]f(x) = f(a).
f(x) is continuous at interval I if it is continuous at all points in the interval.
If f and g are continuous at point a then f±g, fg, f/g (g≠0) are continuous.

Question 26

secx and cscx

Answer 26

1/cosx and 1/sinx respectively

Question 27

e^x

Answer 27

Domain = (-∞, ∞) and Range = (0, ∞)

Question 28

lnx

Answer 28

Domain = (0, ∞) and Range = (–∞, ∞)

Question 29

Domain of gof(x)

Answer 29

{xeD[f] | f(x)eD[g]}

Question 30

Heaviside Formula

Answer 30

When given f(x) = { C1 (-∞, a), x < a ; C2 (a, b), a < x < b ; C3 (b, ∞), x >b }
f(x) = C1[1-H(x-a)] + C2[H(x-a) - H(x-b)] + C3[H(x-b)]