MATH 121 Exam 2 Questions fully solved & verified for accuracy(A+graded)

What is the definition of a derivative?

lim(as h→0) [f(a+h) - f(a)] / [h]

What is the word definition of a derivative?

the slope of the tangent line

What is f'(x) of f(x) = eˣ?

x'∙eˣ

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 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?

f'(x) = cosx

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

f'(x) = -sinx

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

f'(x) = sec²x

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

f'(x) = -csc²x

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

f'(x) = secxtanx

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

f'(x) = -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]

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

[1] / [|x|(√x²-1)]

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

[-1] / [1+x²]

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

- [1] / [|x|(√x²-1)]

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

(1/x)∙x'

What is the derivative of y = bˣ?

bˣ∙lnb

What is the derivative of y = LOGbX?

y' = 1 / xlnb

What MUST you do if you have a function raised to a function?

logarithmic differentiation

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

f'(x) = coshx

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

f'(x) = sinhx

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

f'(x) = sech²x

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

f'(x) = -csch²x

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

f'(x) = -sechxtanhx

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

f'(x) = -cschxcothx

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

[1] / [(√1 + x²)]

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

[1] / [(√x²-1)]

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

[1] / [1-x²]

Steps to figuring out related rates problems

1. read the problem

2. draw a picture and label

3. Write an equation

4. Implicit differentiation

5. Plug-in

6. Answer the questions with the right units

What is the conversion for revolutions?

2𝛑 rad / 1 revolution