MATH 121 Final Exam Questions fully solved & verified for accuracy(A+graded)

How do you solve for | x-c | < R?

1. -R < x-c < R

2. c-R < x < c+R

What is the distance formula?

√(x₂-x₁)² + (y₂-y₁)²

If the center of a circle is at (x₁, y₁) and it has radius r, what is the equation of the circle?

(x-x₁)² + (y-y₁)² = r²

T/F: For a function, for each input there is only one output but one output can have multiple inputs AKA one x can only have one y, but one y can have multiple x values

T

All functions MUST pass the ______ ______ test.

vertical line

What is the domain for all polynomials?

- ∀x (for all x)

What is a rational function? What is the domain of a rational function?

- f(x) = P(x) / Q(x) where P(x) and Q(x) are polynomials

- When Q(x) ≠ 0

What is the domain of an algebraic function (roots)?

when the inside of the root ≥ 0

What is the range of an exponential function?

For f(x) = aˣ, when a > 0

What is the domain for f(x) = sinx and f(x) = cosx?

∀x

What is the domain for tanx?

tanx = sinx / cosx ∴ the domain is anywhere cosx ≠ 0

What do the following translate to?

1. (f+g)(x)

2. (f-g)(x)

3. (fg)(x)

4. (f/g)(x)

1. f(x) + g(x)

2. f(x) - g(x)

3. f(x)g(x)

4. f(x) / g(x)

How is (f∘g)(x) read?

f(g(x))

What does 𝛑 = in degrees?

180°

Using the following image, use the letters to find the following:

1. sinθ

2. cosθ

3. tanθ

4. cotθ

5. secθ

6. cscθ

1. sinθ = b/c

2. cosθ = a/c

3. tanθ = b/a

4. cotθ = a/b

5. secθ = c/a

6. cscθ = c/b

**

Finish the following trig identities:

1. sin²x + cos²x =

2. tan²x + 1 =

3. cot²x + 1 =

4. sin(x+y) =

5. sin(2x) =

6. sin²x =

7. cos²x =

1. 1

2. sec²x

3. csc²x

4. sinxcosy + cosxsiny

5. 2sinxcosx

6. (1-cos2x) / 2

7. (1+cos2x) / 2

How do you know if f(x) and g(x) are inverse functions?

1. If f(g(x)) = x and g(f(x)) = x

2. Must be a one-to-one function AKA f(a)≠f(b) unless a =b

3. Passes the horizontal line test

How do you find the inverse of a function?

1. switch the x and y variables

2. solve for y

What is the inverse of ...

1. f(x) = sinx

2. f(x) = cosx

3. f(x) = tanx

1. arcsinx

2. arccosx

3. arctanx

What is the domain and range of:

1. sinx

2. arcsinx

1. domain: ∀x

range: (-1, 1)

2. domain: (-1, 1)

range: ∀y

What is the domain and range of:

1. cosx

2. arccosx

1. domain: ∀x

range: (-1, 1)

2. domain: (-1, 1)

range: ∀y

Describe the domain and range of (AKA what the graph looks like):

1. tanx

2. arctanx

1.

- There are vertical asymptotes starting at 0 and additions of 𝛑/2 on the x axis

- range goes to infinity

2.

graph of tanx flipped horizontally

(therefore domain is ∀x)

What do the following translate to?

1. bˣbʸ =

2. bˣ/bʸ =

3. (bˣ)ʸ =

1. bˣ⁺ʸ

2. bˣ⁻ʸ

3. bˣʸ

How can you rearrange y = LOGbX?

bʸ = x

What does the following =?

1. LOGb(bˣ)

2. b^(LOGbX)

1. x

2. x

What does the following equal?

1. ln(eˣ) =

2. e^(lnx) =

1. x

2. x

eˣ and lnx are ______

inverses

**

What do the following hyperbolic functions equal?

1. sinhx

2. coshx

1. sinhx = [eˣ - e⁻ˣ] / 2

2. coshx = [eˣ + e⁻ˣ] / 2

What does cosh²x - sinh²x = ?

1

What does the following =?

1. tanhx

2. cothx

1. sinhx / coshx

2. coshx / sinhx

What is average velocity?

Δ distance / Δ time

If you S(t), what does average velocity =?

[S(t₁) - S(t₀)] / t₁ - t₀

What's the definition of a limit?

When x "gets close" to c, f(x) "gets close" to L

What are the 2 rules of limits?

If the lim x→c f(x) = L and lim x→c g(x) = M then . . .

1. lim x→c (f(x) ± g(x)) =

2. lim x→c k(fx) =

3. lim x→c f(x) ∙ g(x) =

4. lim x→c f(x) / g(x) =

5. lim x→c (f(x))ᴺ =

6. lim x→c ᴺ√f(x) =

1. L ± M

2. kL

3. L ∙M

4. L/M , M ≠ 0

5. Lᴺ

6. ᴺ√L , L>0

If the limit of a graph goes to infinity or negative infinity (AKA asymptotes are involved), what are the limits?

DNE

What3 things must be true in order for a function to be continuous?

1. lim x→c f(x) must exist

2. f(c) must exist

3. lim x→c f(x) = f(c)

What are the 2 kinds of discontinuities?

1. removable discontinuity

2. non-removable discontinuity

What is a removable discontinuity?

1. When lim x→c f(x) must exists but lim x→c f(x) ≠ f(c)

What is a non-removable discontinuity?

When there is a jump or an asymptote on the graph

What are one-sided limits?

When the limit as x→c⁻ ≠ x→c⁺, in which case the limit if f(x) in general DNE

If given two piece-wise functions in which one function has an a, and you are told to find a so that f(x) is continuous, how do you find a?

1. Plug in x with the number that x approaches

2. set both answers equal to each other

3. solve for a

*answer: a = 2

Is f(x) ± g(x) continuous?

yes

Is f(x) ∙ g(x) continuous?

yes

Is kf(x) continuous?

yes

When is f(x) / g(x) continuous?

as long as g(x)≠0

Are all polynomials continuous?

yes

Are sinx and cosx continuous?

yes

When is bˣ continuous?

x > 0

Where are tanx and secx continuous?

Where cosx≠0

Where are cotx and cscx continuous?

where sinx≠0

How do you find the limit of a rational function that equals 0 if you plug in c?

factoring and then crossing out like top and bottom

How do you find the limit of a rational function where the numerator contains fractions?

Multiply the numerator's fractions by common factor to get the same denominator

How do you find the limit of a rational function where the numerator involves a root?

Multiply top and bottom by the numerator just with the opposite operation

OR

factor out the denominator so that each factor contains a root

How do you solve the limit of a function that involves trig?

use trig identities to cancel out

What is the lim x→0 [sinx/x] =?

1

How do you use the Squeeze theorem?

1. Break it down to a single trig function

2. Set it -1 ≤ function ≤ 1

3. Make the function look like the original and do what you did to make the function look like the original to the -1 and 1

4. Find the limit of either ends (should be the same)

5. Therefore the function's limit is the same

What is the Squeeze Theorem?

If g(x) ≤ f(x) ≤ h(x) AND lim x→ c g(x) = L, lim x→ c h(x) = LTHENlim x→ c f(x) = L

AKA

If g(x) is going to c and h(x) is going to c, then f(x) has no choice but to go to c w/ them while getting squeezed

What is the lim x→0 [1-cosx]/x = ?

0

How do you know that a limit involves a horizontal asymptote?

When c when x→c is ∞ or -∞

What is the lim x → ±∞ eˣ?

∞: +∞

-∞: 0

What is the lim x → ±∞ sinx?

DNE

What is the lim x → ±∞ cosx?

DNE

1. What is the lim x →+∞ xᴺ?

2. What is the lim x →-∞ xᴺ?

1. +∞

2. +∞ if N is even ; -∞ is N is odd

What is the lim x → ±∞ 1/xᴺ?

0

What is the limit of f(x) = P(x) / Q(x)

1. if the degree of P(x) < degree of Q(x)?

2. if the degree of P(x) > degree of Q(x)?

3. if the degree of P(x) = degree of Q(x)?

1. 0

2. DNE

3. leading coefficient of P(x) / leading coefficient of Q(x)

What should you be aware of when the c as x→c = -∞?

Make the answer the opposite sign of the coefficient that is not under the root

How do you find the limits of functions involving trig functions when the c as x→c = ±∞?

use the squeeze theorem

What is the Intermediate Value Theorem?

If f(x) is continuous on the closed interval [a,b]

AND

f(a) ≠ f(b) then

for any M between f(a) and f(b), there is a c where a < c < b and f(c) = M

How do you use the Intermediate Value theorem?

1. Find f(a)

2. Find f(b)

3. Make sure that M is in between f(a) and f(b)

4. Then the IVT is true

What is the formal definition of a limit?

lim x→c f(x) = L if . . .

for all ε > 0, there exists a δ > 0 such that if 0 < | x-c| < δ

then

|f(x) - L | < ε

How do you solve problems involving δ and ε?

For

lim x→c f(x) = L, ε = y

-You are solving for δ-

1. | f(x) - L | < y

2. Make | f(x) - L | into | x-c | form, applying what you don't need to y

3. Set what is done to L = δ

How do you solve δ and ε problems where you have exponents ≥2 and you need to factor things out?

Same way, except add 1 to the value of c (from x→c) and plug it into the factor that does not take the form of x - c, and then solve for the x - c form

When solving δ and ε problems, when do you plug in the number higher on the number line for x vs. the number lower on the number line?

If there is a quotient then plug in smaller number on the number line; no fraction = greater # on the # line

What is the word definition of a derivative?

the slope of the tangent line

What is the definition of a derivative?

= f'(x) = slope of a tangent line = lim(h→0) [f(x+h) - f(x)] / [h] AKA f'(a) = lim(x→a) f(x)-f(a) / x-a

How do you solve problems with the definition of a derivative?

1. Plug in everything but keep h the same

2. Factor things out so you can somehow cancel out the h in the denominator

3. Plug in 0 for h

All _______ functions are ______ BUT NOT all ______ functions are _______.

- differentiable

- continuous

- continuous

- differentiable

What is an infinite tangent line? Example function?

When as you approach zero, the slope becomes infinity

Ex: x¹⁄³

Is a cusp differentiable?

no, limit DNE

What is the derivative of cf(x)?

cf'(x)

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

= f'(x) ± g'(x)

What is the product rule?

P(x) = f(x)g(x)

f(x)∙g'(x) + g(x)∙f'(x)

What is the quotient rule?

q(x) = f(x) / g(x)

[g(x)∙f'(x) - f(x)∙g'(x)] / [g(x)]²

What is the reciprocal rule?

f(x) = 1 / g(x)

[-g'(x)] / [g(x)]²

What is the average rate of change?

(x₀, f(x₀)) , (x₁, f(x₁))

Δy / Δx = [f(x₁) - f(x₂)] / [x₁ - x₀]

What is the instantaneous rate of change?[x₀, f(x₀)] , [x₁, f(x₁)]

lim [Δx → 0] Δy / Δx = f'(x₀)

What is the equation for height?

s = s₀ + v₀t - 0.5gt²

s₀ = initial height

v₀ = initial velocity

g = acceleration of gravity

What does g = ?

9.8 m/s² OR 32 ft/s²

What is the equation for velocity?

v = ds/dt = v₀ - gt

**

What is the derivative of f(x) = sinx?

cosx

**

What is the derivative of f(x) = cosx?

-sinx

**

What is the derivative of f(x) = tanx?

sec²x

**

What is the derivative of f(x) = secx?

secxtanx

**

What is the derivative of f(x) = cotx?

-csc²x

**

What is the derivative of f(x) = cscx?

-cscxcotx

What is the chain rule?

q(x) = f(g(x))

q'(x) = f'(g(x))∙g'(x)

**

What is the derivative of f(x) = arcsinx?

[1] / [(√1-x²)]

*

What is the derivative of f(x) = arccosx?

[-1] / [(√1-x²)]

**

What is the derivative of f(x) = arctanx?

[1] / [1 + x²]