unit-2 – HighFiveAP

2.1

Average vs. Instantaneous Rate Concept Core

Average rate uses two points (secant). Instantaneous rate is the slope at one point (tangent) — that’s the derivative.

Two Slopes You Must Distinguish

📉 Average Rate (Secant)

Slope from x=a to x=b.

f(b)−f(a)b−a

🎯 Instantaneous Rate (Tangent)

Slope exactly at x=c.

f'(c)= limh→0 f(c+h)−f(c)h

AP Exam Moves

Average rate on [a,b] is secant slope. Instantaneous at x=c is tangent slope.
Units: if s is meters and t is seconds, then s'(t) is meters/second.

Common Mistake

Using f'(a) when the question asks for average rate on [a,b]. Two points → secant.

Mini Practice

1) If f(2)=5 and f(6)=13, find average rate on [2,6].

2) If f'(4)=7, what does 7 represent?

Show Answers

1) (13−5)/(6−2)=8/4=2 .

2) The slope of the tangent line at x=4 (instantaneous rate of change at 4).

2.2 - 2.3

The Definition of the Derivative FRQ Setup

The derivative is defined as a limit. On AP, you must be able to (1) set it up correctly and (2) simplify cleanly.

Limit Definition (Memorize)

f'(x)= limh→0 f(x+h)−f(x)h

Think: “difference quotient” as h shrinks to 0.

f'(a)= limx→a f(x)−f(a)x−a

Same idea, different variable. Useful for “derivative at a.”

Derivative Notation (AP uses ALL of these)

Meaning	Common Notation
Derivative of y with respect to x	y', f'(x), d/dx[f(x)], dy/dx
Derivative evaluated at a point	f'(a), (dy/dx)\|x=a

AP “Do it clean” example: f(x)=x²

Setup: limh→0 [(x+h)²−x²]/h
Expand: limh→0 [x²+2xh+h²−x²]/h
Simplify: limh→0 [2xh+h²]/h
Factor: limh→0 h(2x+h)/h
Cancel: limh→0 (2x+h)=2x

Common Mistake

Dropping parentheses: it must be f(x+h)−f(x).

Mini Practice

1) Use the definition to find f'(x) for f(x)=3x+4.

Show Answer

f'(x)= lim h→0 [3(x+h)+4−(3x+4)]/h = limh→0 (3h)/h = 3 .

2.4

Differentiability DNE Patterns

Differentiable means there is a unique tangent slope at that point. If the graph has a “problem point,” then f'(c) does not exist.

The One Rule to Memorize

Differentiable at c ⇒ Continuous at c

If it’s not continuous, it cannot be differentiable.

How Differentiability Fails (AP favorites)

Discontinuity
hole / jump / asymptote

Corner/Cusp
left slope ≠ right slope

Vertical Tangent
slope → ±∞

Common Mistake

Saying “continuous so differentiable.” False (|x| at 0).

2.5 - 2.7

Derivative Rules + Linearity No-Calc Speed

Most derivatives are computed using rules. To score a 5, your algebra must be fast and your rule use must be automatic.

Linearity Rules

d/dx [c·f(x)] = c·f'(x) and d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Core Derivative Table (Unit 2)

Function	Derivative	Notes
xn	n·xn−1	Power rule
c	0	Constant → 0
ex	ex	Same function
ln(x)	1/x	Domain: x>0
sin(x)	cos(x)
cos(x)	−sin(x)	Negative appears here

2.8 - 2.9

Product & Quotient Rules FRQ Mechanics

Product Rule

(fg)' = f'g + fg'

Quotient Rule

(f/g)' = g f' − f g'g2

2.10

Other Trig Derivatives Must Memorize

Trig Derivative Pairs

tan(x) → sec²(x)

cot(x) → −csc²(x)

sec(x) → sec(x)tan(x)

csc(x) → −csc(x)cot(x)

Unit 2

Everything You Need for a 5 Checklist

If you can do ALL of these quickly + accurately, Unit 2 is locked.

Average vs instantaneous rate (secant vs tangent) + units.
Derivative definition set-up (both forms) + clean simplification.
Derivative notation fluency (f'(x), dy/dx, d/dx).
When derivative DNE: discontinuity, corner/cusp, vertical tangent.
Power rule + linearity without algebra mistakes.
Product/quotient rules done correctly (simplify first when smart).
Trig + ex + ln derivative memorized.