How to Use This Problem Set
Work each problem before reading the solution. Cover the answer, attempt the problem fully, then compare your method. Pay attention to the technique used — not just the answer. The goal is to build pattern recognition so you automatically know which method applies.
Limits (Problems 1–10)
Derivatives (Problems 11–25)
Integrals (Problems 26–40)
Applications (Problems 41–50)
Work each problem independently before looking at the solution. The goal is not to verify answers but to build pattern recognition. If you needed to look at the solution, redo a similar problem from scratch after reviewing the technique.
The Science of Practice in Mathematics
Mathematical skill is built through what researchers call "deliberate practice" — structured, effortful repetition with immediate feedback. For calculus specifically, this means: identify the technique (not just execute it), attempt the problem fully, compare your approach to the model solution, identify exactly where any errors occurred, and immediately attempt a similar problem to consolidate the correction. Passive re-reading of solutions does not build skill; active problem-solving does.
Research in mathematics education consistently shows that "spaced practice" — returning to problems after a delay — produces better long-term retention than massed practice (doing all problems on one topic in one session). After completing the limit problems, move to derivatives, return to limits after a few days, then test yourself again. This interleaving feels harder but produces superior retention.
Limits — 10 Problems with Full Solutions
P1: lim(x→4) (x²−16)/(x−4). Answer: Factor (x+4)(x−4)/(x−4) = x+4 → 8.
P2: lim(x→0) sin(3x)/(5x). Answer: (3/5)·lim sin(3x)/(3x) = 3/5.
P3: lim(x→∞) (2x³−x)/(x³+7). Answer: Divide by x³: (2−1/x²)/(1+7/x³) → 2.
P4: lim(x→1) (x³−1)/(x²−1). Answer: Factor: (x−1)(x²+x+1)/[(x−1)(x+1)] = (x²+x+1)/(x+1) → 3/2.
P5: lim(x→0) (1−cos x)/x². Answer: L'Hôpital twice: sin x/(2x) → cos x/2 → 1/2.
P6: lim(x→∞) x·sin(1/x). Answer: Let t=1/x → lim(t→0) sin(t)/t = 1.
P7: lim(x→2⁺) 1/(x−2). Answer: As x→2 from right, x−2→0⁺, so 1/(x−2)→+∞.
P8: lim(x→0) (eˣ−1−x)/x². Answer: Taylor series: (x+x²/2+···−1−(1+x−1))/x² → Wait: eˣ−1−x = x²/2+x³/6+··· Divide by x²: → 1/2.
P9: lim(x→π/2) tan(x). Answer: As x→π/2, cos x→0 so tan x = sin x/cos x → +∞ (from left) or −∞ (from right). Limit DNE (one-sided limits differ in sign).
P10: lim(x→0) x·ln(x). Answer: Rewrite as ln(x)/(1/x), apply L'Hôpital: (1/x)/(−1/x²) = −x → 0.
Derivatives — 15 Problems with Solutions
P11: d/dx[x⁶−4x³+2x−7] = 6x⁵−12x²+2.
P12: d/dx[sin(x²+1)] = cos(x²+1)·2x = 2x·cos(x²+1).
P13: d/dx[eˣ·ln x] = eˣ·ln x + eˣ·(1/x) = eˣ(ln x + 1/x).
P14: d/dx[(x³+1)/(x²−1)] = [3x²(x²−1)−(x³+1)·2x]/(x²−1)² = [x⁴−3x²−2x]/(x²−1)².
P15: dy/dx if x²+xy+y²=7. Implicit: 2x+y+x(dy/dx)+2y(dy/dx)=0 → dy/dx = −(2x+y)/(x+2y).
P16: d/dx[arctan(2x)] = 1/(1+(2x)²)·2 = 2/(1+4x²).
P17: d/dx[(3x+1)⁵] = 5(3x+1)⁴·3 = 15(3x+1)⁴.
P18: d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x).
P19: If f(x) = ln(cos x), find f'(x). = (−sin x)/cos x = −tan x.
P20: d²y/dx² if y = x⁴−3x². y' = 4x³−6x. y'' = 12x²−6.
P21: Find the tangent line to y = √x at x = 4. y(4) = 2. y'(x) = 1/(2√x). y'(4) = 1/4. Tangent: y−2 = (1/4)(x−4) → y = x/4+1.
P22: d/dx[e^(sin x)] = e^(sin x)·cos x.
P23: d/dx[xˣ] = d/dx[e^(x ln x)] = e^(x ln x)·(ln x + 1) = xˣ(ln x + 1).
P24: Related rate: sphere radius growing at 2 cm/s. Rate of volume change when r = 5. V = (4/3)πr³. dV/dt = 4πr²·(dr/dt) = 4π(25)(2) = 200π cm³/s.
P25: Find critical points of f(x) = x³−9x+5. f'(x) = 3x²−9 = 0 → x² = 3 → x = ±√3. f''(√3) = 6√3 > 0 (min). f''(−√3) = −6√3 < 0 (max).
- Stewart, J. (2015). Calculus, End-of-chapter exercises. Cengage.
- Larson, R. & Edwards, B. (2013). Calculus, 10th ed. Cengage.
- College Board (2024). AP Calculus AB Free-Response Questions. collegeboard.org.