πŸ“ Calculus AB
Self-Study Notebook Β· 20 Core Problems
✏️ Write, Think, Learn
✏️ Progress
0 / 20 βœ“
Unit 1 Limits & Continuity p. 1
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πŸ”‘ Quick Memory Point
APPROACH β€” Limit = what value f(x) approaches, NOT the value AT x.
LEFT β‰  RIGHT β€” If lim from left β‰  lim from right β†’ limit DNE
HOLE vs VERTICAL β€” 0/0 β†’ factor & cancel (removable) Β· c/0 β†’ vertical asymptote
πŸ“– Worked Example
Find \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\)
  1. Direct sub: \(\frac{9-9}{3-3} = \frac{0}{0}\) β†’ indeterminate β†’ factor!
  2. Factor: \(\frac{(x-3)(x+3)}{x-3} = x+3\)
  3. Now sub: \(3 + 3 = \mathbf{6}\) βœ“
βœ“ CORRECT!
1
Evaluate: \(\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)
⚠️ Common trap: Don't just plug in 2 directly!
βœ“ CORRECT!
2
Given \(f(x) = \begin{cases} x + 1 & x < 2 \\ 5 & x = 2 \\ 3x - 2 & x > 2 \end{cases}\)

Which statement is TRUE?
βœ“ CORRECT!
3
\(\displaystyle\lim_{x \to 0} \frac{\sin(3x)}{x} = \) ?
✏️ Hint: recall the fundamental trig limit \(\lim_{x\to 0}\frac{\sin x}{x}=1\)
βœ“ CORRECT!
4
What is \(\displaystyle\lim_{x \to \infty} \frac{4x^2 - 3x}{2x^2 + 7}\)?
⚠️ Trick: compare leading terms only!
✏️ My notes:
Unit 2–3 Derivatives p. 2
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πŸ”‘ Quick Memory Point
POWER RULE: \(x^n \to nx^{n-1}\) β€” "bring it down, reduce by 1"
CHAIN: outside' Β· inside' β€” always work outside-in
PRODUCT: \((uv)' = u'v + uv'\) β€” "firstΒ·d(second) + secondΒ·d(first)"
QUOTIENT: LO-d(HI) minus HI-d(LO) over LOΒ·LO β€” "Lo dee Hi minus Hi dee Lo, over Lo Lo go!"
πŸ“– Worked Example β€” Chain Rule
Differentiate \(y = \sin(x^2 + 1)\)
  1. Outer function: \(\sin(\square)\), outer derivative = \(\cos(\square)\)
  2. Inner function: \(x^2+1\), inner derivative = \(2x\)
  3. Chain: \(y' = \cos(x^2+1) \cdot 2x\)
βœ“ CORRECT!
5
If \(f(x) = 3x^4 - 2x^3 + 5x - 7\), find \(f'(x)\).
βœ“ CORRECT!
6
Find \(\dfrac{dy}{dx}\) for \(y = (3x^2 + 1)^5\)
⚠️ Don't forget the inner derivative!
βœ“ CORRECT!
7
Differentiate \(y = x^2 \cdot e^x\). Use the Product Rule.
βœ“ CORRECT!
8
Find \(\dfrac{dy}{dx}\) using implicit differentiation: \(x^2 + y^2 = 25\)
⚠️ Don't forget: \(\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}\)
✏️ My notes:
Unit 4–5 Applications of Derivatives p. 3
⚑
πŸ”‘ Quick Memory Point
INCREASING: f'(x) > 0 Β· DECREASING: f'(x) < 0
CRITICAL PT: f'(x) = 0 or DNE β†’ candidate for max/min
CONCAVE UP: f''(x) > 0 (smile 😊) · CONCAVE DOWN: f''(x) < 0 (frown 😞)
INFLECTION: f'' changes sign β†’ shape changes
πŸ“– Worked Example β€” First Derivative Test
\(f(x) = x^3 - 3x\) β€” find local extrema
  1. \(f'(x) = 3x^2 - 3 = 3(x-1)(x+1)\) β†’ critical pts: \(x = \pm 1\)
  2. Sign chart: \(f'(-2)=+\), \(f'(0)=-\), \(f'(2)=+\)
  3. x = βˆ’1: + β†’ βˆ’ β†’ local max; x = 1: βˆ’ β†’ + β†’ local min
βœ“ CORRECT!
9
\(f(x) = 2x^3 - 9x^2 + 12x\). On which interval is \(f\) increasing?
βœ“ CORRECT!
10
A particle moves along a line with velocity \(v(t) = t^2 - 4t + 3\). When is the particle moving left (in the negative direction)?
βœ“ CORRECT!
11
The Mean Value Theorem guarantees a point \(c\) on \([1, 4]\) where \(f'(c) = \dfrac{f(4)-f(1)}{4-1}\) if \(f\) is:
βœ“ CORRECT!
12
A 10 ft ladder leans against a wall. If the bottom slides away at 2 ft/sec, how fast is the top sliding down when the bottom is 6 ft from the wall?
⚠️ Related rates β€” draw a picture first!
✏️ My notes:
Unit 6 Integration p. 4
⚑
πŸ”‘ Quick Memory Point
POWER RULE ∫: \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) β€” "add 1, divide by new power"
+C ALWAYS! Indefinite integral β†’ never forget \(+C\)
U-SUB: let u = inside β†’ du replaces messy part β†’ integrate β†’ back-sub
FTC1: \(\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\) β€” derivative of integral = integrand!
πŸ“– Worked Example β€” u-Substitution
\(\displaystyle\int 2x\,e^{x^2}\,dx\)
  1. Let \(u = x^2\), so \(du = 2x\,dx\)
  2. Substitute: \(\displaystyle\int e^u\,du = e^u + C\)
  3. Back-sub: \(e^{x^2} + C\) βœ“
βœ“ CORRECT!
13
Evaluate \(\displaystyle\int (4x^3 - 6x + 2)\,dx\)
βœ“ CORRECT!
14
\(\displaystyle\int_0^3 (x^2 + 1)\,dx = \) ?
⚠️ Definite integral β†’ No +C needed!
βœ“ CORRECT!
15
Use u-substitution: \(\displaystyle\int 3x^2\cos(x^3)\,dx\)
βœ“ CORRECT!
16
Which property is always true?
\(\displaystyle\int_a^b f(x)\,dx = \text{ ?}\)
✏️ My notes:
Unit 7–8 FTC, Area, & Volume p. 5
⚑
πŸ”‘ Quick Memory Point
AREA between curves: \(\int_a^b [top - bottom]\,dx\) β€” always TOP minus BOTTOM
DISK method: \(V = \pi\int_a^b [f(x)]^2\,dx\) β€” rotating around x-axis
WASHER: \(V = \pi\int [R^2 - r^2]\,dx\) β€” outer radiusΒ² βˆ’ inner radiusΒ²
AVG VALUE: \(\frac{1}{b-a}\int_a^b f(x)\,dx\)
πŸ“– Worked Example β€” Area Between Curves
Area between \(y = x^2\) and \(y = x\) on \([0,1]\):
  1. Identify top: \(x \geq x^2\) on [0,1] β†’ top = \(x\), bottom = \(x^2\)
  2. \(\displaystyle\int_0^1(x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1\)
  3. \(= \frac{1}{2} - \frac{1}{3} = \mathbf{\frac{1}{6}}\)
βœ“ CORRECT!
17
If \(\displaystyle F(x) = \int_1^x \sqrt{t^2+1}\,dt\), then \(F'(x) = \) ?
⚠️ This is the Fundamental Theorem of Calculus Part 1!
βœ“ CORRECT!
18
Find the average value of \(f(x) = x^2\) on the interval \([0, 3]\).
βœ“ CORRECT!
19
The area of the region bounded by \(y = x^2\) and \(y = 4\) is:
βœ“ CORRECT!
20
The region bounded by \(y = \sqrt{x}\), \(x = 0\), and \(x = 4\) is rotated about the x-axis. Volume = ?
⚠️ Disk method: \(V = \pi\int_a^b [f(x)]^2\,dx\)
✏️ Final notes & formulas I want to remember: