AP Calculus AB — Core Problems

Q01

Evaluate the limit: \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\)

Easy

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Memory Point

FACTOR & CANCEL — If you get 0/0, always try to factor numerator. Cancel the common factor, then plug in.

Quick Example \(\frac{x^2-9}{x-3} = \frac{(x+3)(x-3)}{x-3} = x+3\) → plug in \(x=3\): answer is \(6\).

A\(0\)

B\(6\)

C\(3\)

DDoes not exist

Q02

Which condition is NOT required for \(f\) to be continuous at \(x = a\)?

Easy

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Memory Point

3-CHECK RULE: (1) \(f(a)\) exists, (2) \(\lim_{x\to a}f(x)\) exists, (3) they are equal. All three must hold.

A\(f(a)\) is defined

B\(\displaystyle\lim_{x \to a} f(x)\) exists

C\(f\) is differentiable at \(x = a\)

D\(\displaystyle\lim_{x \to a} f(x) = f(a)\)

Q03

Find \(\displaystyle\lim_{x \to \infty} \frac{5x^3 - 2x}{3x^3 + 7}\)

Easy

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Memory Point

LEADING COEFFICIENTS — Same degree on top & bottom → divide the leading coefficients. \(\frac{5}{3}\) is the answer directly.

A\(\dfrac{5}{3}\)

B\(0\)

C\(\infty\)

D\(\dfrac{5}{7}\)

Q04

If \(f(x) = x^4 - 3x^2 + 5x - 2\), find \(f'(x)\).

Easy

⚡

Memory Point

POWER RULE: \(\frac{d}{dx}[x^n] = nx^{n-1}\). Bring exponent down, subtract 1. Constants vanish.

A\(4x^3 - 6x + 5\)

B\(4x^4 - 6x^2 + 5\)

C\(4x^3 - 6x + 5\)

D\(4x^3 - 3x + 5\)

Q05

Find \(\dfrac{d}{dx}[\sin(x^2)]\).

Easy

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Memory Point

CHAIN RULE: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\). Outside derivative × inside derivative.

A\(\cos(x^2)\)

B\(2x\cos(x^2)\)

C\(2x\sin(x^2)\)

D\(-2x\cos(x^2)\)

Q06

Find \(\dfrac{d}{dx}\!\left[\dfrac{x^2}{e^x}\right]\).

Medium

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Memory Point

QUOTIENT RULE: \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\). Remember: "low d-high minus high d-low, over low squared."

A\(\dfrac{2x}{e^x}\)

B\(\dfrac{x^2 - 2x}{e^x}\)

C\(\dfrac{x^2 + 2x}{e^{2x}}\)

D\(\dfrac{2x - x^2}{e^x}\)

Q07

If \(y = \ln(3x^2 + 1)\), find \(y'\).

Easy

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Memory Point

ln CHAIN: \(\frac{d}{dx}[\ln u] = \frac{u'}{u}\). Derivative of inside over inside.

A\(\dfrac{6x}{3x^2+1}\)

B\(\dfrac{1}{3x^2+1}\)

C\(\dfrac{6x}{(3x^2+1)^2}\)

D\(\ln(6x)\)

Q08

For \(f(x) = 2x^3 - 9x^2 + 12x\), on which interval is \(f\) increasing?

Medium

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Memory Point

f' > 0 → INCREASING. Find critical points where \(f'=0\), then test sign of \(f'\) in each interval.

Setup \(f'(x) = 6x^2 - 18x + 12 = 6(x-1)(x-2)\). Critical points: \(x=1\) and \(x=2\).

A\((1,\, 2)\)

B\((-\infty,1)\) and \((2,\infty)\)

C\((0,\, 3)\)

D\((-\infty,\, \infty)\)

Q09

At \(x = 2\), the function \(f(x) = x^3 - 6x^2 + 9x\) has a:

Medium

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Memory Point

2nd DERIVATIVE TEST: \(f''>0\) → local min, \(f''<0\) → local max, \(f''=0\) → inconclusive. Check \(f'=0\) first!

Setup \(f'(x)=3x^2-12x+9=3(x-1)(x-3)\). At \(x=2\): \(f'(2)=3(1)(-1)=-3\neq 0\). Not a critical point.

ALocal maximum

BLocal minimum

CNeither — \(x=2\) is not a critical point

DInflection point

Q10

The Mean Value Theorem guarantees that if \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c\) in \((a,b)\) such that:

Easy

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Memory Point

MVT = SLOPE MATCH: Instantaneous rate equals average rate somewhere. "Speeding ticket theorem."

A\(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)

B\(f(c) = \dfrac{f(a)+f(b)}{2}\)

C\(f'(c) = f(b) - f(a)\)

D\(f''(c) = 0\)

Q11

Use implicit differentiation: if \(x^2 + y^2 = 25\), find \(\dfrac{dy}{dx}\).

Medium

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Memory Point

IMPLICIT: Differentiate both sides w.r.t. \(x\). Every \(y\)-term gets a \(\frac{dy}{dx}\) multiplied. Then solve for \(\frac{dy}{dx}\).

A\(\dfrac{y}{x}\)

B\(-\dfrac{x}{y}\)

C\(\dfrac{x}{y}\)

D\(-\dfrac{y}{x}\)

Q12

Evaluate \(\displaystyle\int (4x^3 - 2x + 7)\, dx\).

Easy

⚡

Memory Point

REVERSE POWER RULE: \(\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\). Add 1 to exponent, divide by new exponent. NEVER forget \(+C\).

A\(12x^2 - 2 + C\)

B\(x^4 - x^2 + 7x\)

C\(x^4 - x^2 + 7x + C\)

D\(4x^4 - 2x^2 + 7x + C\)

Q13

Evaluate \(\displaystyle\int_0^2 (3x^2 + 1)\, dx\).

Easy

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Memory Point

FTC Part 2: \(\int_a^b f(x)\,dx = F(b)-F(a)\). Find antiderivative, plug in top limit minus bottom limit.

A\(10\)

B\(8\)

C\(12\)

D\(14\)

Q14

Evaluate \(\displaystyle\int 2x\cos(x^2)\, dx\).

Medium

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Memory Point

U-SUB SIGNAL: See a function and its derivative next to each other → let \(u\) = inner function. Here \(u=x^2\), \(du=2x\,dx\). ✓

A\(-\sin(x^2)+C\)

B\(\sin(x^2)+C\)

C\(2\sin(x^2)+C\)

D\(\cos(x^2)+C\)

Q15

Let \(g(x) = \displaystyle\int_0^x \sqrt{t^2+1}\, dt\). Find \(g'(x)\).

Easy

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Memory Point

FTC Part 1: \(\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\). Just swap \(t\) for \(x\). That's it — no integration needed!

A\(\displaystyle\int_0^x \frac{t}{\sqrt{t^2+1}}\,dt\)

B\(\sqrt{x^2+1} - 1\)

C\(\dfrac{x}{\sqrt{x^2+1}}\)

D\(\sqrt{x^2+1}\)

Q16

Find the area between \(f(x) = x^2\) and \(g(x) = x\) from \(x=0\) to \(x=1\).

Medium

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Memory Point

TOP MINUS BOTTOM: Area = \(\int_a^b [\text{top} - \text{bottom}]\,dx\). Check which function is on top first! Here \(g(x)=x\geq x^2=f(x)\) on \([0,1]\).

A\(\dfrac{1}{3}\)

B\(\dfrac{1}{4}\)

C\(\dfrac{1}{6}\)

D\(\dfrac{1}{2}\)

Q17

A particle moves with velocity \(v(t) = t^2 - 4\). What is the displacement from \(t=0\) to \(t=3\)?

Medium

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Memory Point

DISPLACEMENT vs DISTANCE: Displacement = \(\int v(t)\,dt\) (can be negative). Distance = \(\int|v(t)|\,dt\) (always positive). Different!

A\(-3\)

B\(3\)

C\(0\)

D\(9\)

Q18

Which statement is always true?

Medium ★ Tricky

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Memory Point

DIFF → CONT, NOT REVERSE: Differentiable ⟹ Continuous. But Continuous ⟹ NOT necessarily differentiable. (e.g., \(|x|\) is continuous but not differentiable at 0.)

AIf \(f\) is continuous at \(x=a\), then \(f\) is differentiable at \(x=a\).

BIf \(f\) is differentiable at \(x=a\), then \(f\) is continuous at \(x=a\).

CIf \(f'(a) = 0\), then \(f\) has a local extremum at \(x=a\).

DIf \(f''(a) = 0\), then \(f\) has an inflection point at \(x=a\).

Q19

A sphere's radius increases at \(2\) cm/s. How fast is the volume increasing when \(r = 3\) cm? (\(V = \tfrac{4}{3}\pi r^3\))

Medium

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Memory Point

RELATED RATES STEPS: (1) Write equation, (2) differentiate both sides w.r.t. \(t\), (3) plug in known values, (4) solve for unknown rate.

A\(24\pi\) cm³/s

B\(36\pi\) cm³/s

C\(72\pi\) cm³/s

D\(18\pi\) cm³/s

Q20

Given \(\displaystyle\int_1^5 f(x)\,dx = 12\) and \(\displaystyle\int_1^3 f(x)\,dx = 5\), find \(\displaystyle\int_3^5 f(x)\,dx\).

Easy ★ Often Missed

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Memory Point

ADDITIVE INTERVALS: \(\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx\). Think of it like adding segments of a number line.

A\(5\)

B\(17\)

C\(60\)

D\(7\)