Calculus I - Week 12

Final Review: Calculus I

This final week synthesizes all of Calculus I. Students revisit the key ideas — limits, derivatives, and integrals — as a unified system, work mixed-topic problems, and practice the kinds of questions that appear on exams and physics courses. No new concepts; maximum depth on what you have learned.

Learning Objectives

- Evaluate limits using all learned techniques. - Differentiate any function using power, product, quotient, chain, and trig rules. - Apply derivatives to optimization, related rates, and curve sketching. - Evaluate definite and indefinite integrals. - Apply the Fundamental Theorem of Calculus in both directions. - Solve initial value problems. - Recognize which technique to use from problem context alone.

1. The Big Picture: What Calculus I Is About

Calculus I has three pillars: 1. LIMITS — the language of calculus. All other concepts are defined using limits. 2. DERIVATIVES — instantaneous rate of change. The slope of a curve at a point. Used for optimization, related rates, linear approximation, and motion. 3. INTEGRALS — accumulated change. Area under a curve. The Fundamental Theorem links derivatives and integrals. Every problem you solved this semester is an application of one or more of these three ideas.

2. Limits Review

Techniques you should know: 1. Direct substitution (try first). 2. Factoring (for 0/0 forms). 3. Conjugates (for radicals). 4. Special limits: lim x->0 sin(x)/x = 1. 5. One-sided limits and when two-sided limits fail. 6. Continuity: a function is continuous at a when lim = f(a). Common errors: - Saying 0/0 = 0 or undefined without doing more work. - Forgetting to check one-sided limits for piecewise functions.

3. Derivative Rules Review

Power rule: d/dx[x^n] = nx^(n-1) Constant: d/dx[c] = 0 Sum/difference: differentiate term by term Product: (fg)' = f'g + fg' Quotient: (f/g)' = (f'g - fg')/g^2 Chain: d/dx[f(g(x))] = f'(g(x)) * g'(x) Trig: d/dx[sin] = cos d/dx[cos] = -sin d/dx[tan] = sec^2 Implicit: treat y as a function of x, write dy/dx when you see a y term.

4. Applications of Derivatives Review

OPTIMIZATION: - Identify objective function and constraint. - Express objective in one variable. - Find critical points, verify with second derivative test. RELATED RATES: - Write the geometric relation. - Differentiate with respect to t. - Substitute known values (after differentiating!). CURVE SKETCHING: - Critical points from f' = 0. - Sign chart for f' (increasing/decreasing). - Sign chart for f'' (concave up/down). - Inflection points where f'' changes sign. LINEAR APPROXIMATION: - L(x) = f(a) + f'(a)(x - a). - Works best when x is near a.

5. Integration Review

INDEFINITE INTEGRALS: ∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1) ∫ cos(x) dx = sin(x) + C ∫ sin(x) dx = -cos(x) + C ∫ sec^2(x) dx = tan(x) + C DEFINITE INTEGRALS (FTC Part 2): ∫[a to b] f(x) dx = F(b) - F(a) FTC PART 1: d/dx [∫[a to x] f(t) dt] = f(x) INITIAL VALUE: Integrate, then use the given condition to find C.

6. Mixed Practice: Limits

Problem A: Evaluate lim x->3 (x^2 - 9)/(x - 3). Factor: (x-3)(x+3)/(x-3) = x+3 → answer: 6. Problem B: Evaluate lim x->0 sin(3x)/(2x). Rewrite: (3/2) * sin(3x)/(3x) → (3/2)*1 = 3/2. Problem C: Evaluate lim x->4 (sqrt(x) - 2)/(x - 4). Conjugate: 1/(sqrt(x)+2) → at x=4: 1/4.

7. Mixed Practice: Derivatives

Problem A: Differentiate f(x) = x^3 * sin(x). Product rule: 3x^2 sin(x) + x^3 cos(x). Problem B: Differentiate f(x) = (x^2 + 1)^5 / (x^3 + x). Quotient rule + chain rule in numerator. Problem C: Find dy/dx for x^2 + xy + y^2 = 7. Implicit: 2x + (y + x dy/dx) + 2y dy/dx = 0. dy/dx = -(2x + y)/(x + 2y).

8. Mixed Practice: Optimization and Related Rates

Problem A: Minimize cost of a box (volume 64) with square base, open top, if base costs $2 and sides cost $1 each. Solve: x = 4, h = 4, min cost = $96. Problem B: A ladder slides. At x = 3, y = 4 in a right triangle with hypotenuse 5. If dx/dt = 2, find dy/dt. Answer: dy/dt = -3/2. Problem C: Find the maximum rectangle inscribed in y = 4 - x^2 above the x-axis. Solve: x = 2/sqrt(3), max area = 16sqrt(3)/9 * 2... or just set A = 2x(4-x^2) and differentiate.

9. Mixed Practice: Integrals

Problem A: ∫[0 to 3] (x^2 - 2x + 1) dx = [x^3/3 - x^2 + x] from 0 to 3 = 9 - 9 + 3 = 3. Problem B: v(t) = 3t^2 - 6t, v(0) = 0, s(0) = 5. Find s(t). s = ∫v dt = t^3 - 3t^2 + C. C=5. s(t) = t^3 - 3t^2 + 5. Problem C: Find area between y = x^2 and y = 2x. Intersect at x=0 and x=2. Area = ∫[0,2](2x-x^2)dx = [x^2 - x^3/3] from 0 to 2 = 4 - 8/3 = 4/3.

10. Exam Strategy

1. For limits: try direct substitution first. If 0/0, try factoring or conjugates. 2. For derivatives: identify what rule(s) are needed before computing. Product → two-term answer. Chain → multiply by inside derivative. 3. For optimization: write the equation you want to optimize in one variable before differentiating. 4. For integrals: check your antiderivative by differentiating it. 5. For FTC: distinguish Part 1 (differentiate an integral) from Part 2 (evaluate with antiderivative). 6. Show your work — partial credit comes from correct setup even if arithmetic slips.