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Day 17 · Review + Quiz

Phase 2 Final Review

Every formula from Day 12–16 in one place. Skim it, then take the quiz below when you're ready.

Extreme Value Theorem & Critical Points (Day 12)Mean Value Theorem & Rolle's Theorem (Day 13)Increasing/Decreasing, First & Second Derivative Tests, Concavity (Day 14)L'Hôpital's Rule (Day 15)Optimization (Day 16)

Extreme Value Theorem & Critical Points (Day 12)Forgot it? Go back and study again →

Critical point

f′(c)=0f'(c)=0 or f′(c)f'(c) undefined

Local extrema can ONLY occur at critical points or endpoints — but not every critical point is one.

Extreme Value Theorem

ff continuous on [a,b][a,b]   ⟹  \implies ff attains an absolute max and min on [a,b][a,b]

Closed interval method

Evaluate ff at all critical points in (a,b)(a,b) plus both endpoints a,ba,b; compare.

Mean Value Theorem & Rolle's Theorem (Day 13)Forgot it? Go back and study again →

Rolle's Theorem

ff continuous on [a,b][a,b], differentiable on (a,b)(a,b), f(a)=f(b)f(a)=f(b)   ⟹  \implies ∃ c∈(a,b):f′(c)=0\exists\, c \in (a,b): f'(c)=0

Mean Value Theorem

ff continuous on [a,b][a,b], differentiable on (a,b)(a,b)   ⟹  \implies ∃ c∈(a,b):f′(c)=f(b)−f(a)b−a\exists\, c \in (a,b): f'(c) = \frac{f(b)-f(a)}{b-a}

Increasing/Decreasing, First & Second Derivative Tests, Concavity (Day 14)Forgot it? Go back and study again →

Monotonicity

f′(x)>0  ⟹  f'(x)>0 \implies increasing. f′(x)<0  ⟹  f'(x)<0 \implies decreasing.

First Derivative Test

At critical point cc: f′f' changes −→+-\to+   ⟹  \implies local min. +→−+\to-   ⟹  \implies local max. No change   ⟹  \implies neither.

Concavity

f′′(x)>0  ⟹  f''(x)>0 \implies concave up. f′′(x)<0  ⟹  f''(x)<0 \implies concave down.

Inflection point

Concavity actually changes at cc (requires f′′(c)=0f''(c)=0 or undefined)

Second Derivative Test

At f′(c)=0f'(c)=0: f′′(c)>0  ⟹  f''(c)>0 \implies local min. f′′(c)<0  ⟹  f''(c)<0 \implies local max. f′′(c)=0  ⟹  f''(c)=0 \implies inconclusive (use First Derivative Test).

L'Hôpital's Rule (Day 15)Forgot it? Go back and study again →

Basic L'Hôpital's Rule

00\frac{0}{0} or ∞∞\frac{\infty}{\infty}: lim⁡f(x)g(x)=lim⁡f′(x)g′(x)\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}

Differentiate top and bottom separately — never the quotient rule. Repeat if still indeterminate.

Other indeterminate forms

∞⋅0\infty\cdot 0 and ∞−∞\infty-\infty: combine into a single fraction first. Exponential forms (00,∞0,1∞0^0,\infty^0,1^\infty): take ln⁡\ln, evaluate, then exponentiate back with e(⋅)e^{(\cdot)}.

Optimization (Day 16)Forgot it? Go back and study again →

4-step strategy

1. Draw & label. 2. Write the primary equation (2 variables). 3. Write the constraint, solve for one variable, substitute. 4. Find the absolute extremum.

Timed Multiple Choice Quiz

Phase 2 Final Quiz: Applications of the Derivative

25 timed questions covering everything from Day 12–16. Treat it like a real exam — review your mistakes after.

45 min

Duration

25

Questions

Before you begin

·Pick one answer per question — you can change it any time before submitting.

·The timer starts immediately when you click Start.

·You can submit early at any time.

·Correct answers and explanations only appear after you submit.