MTH 102 Test Compilation 5 1

Calculus Study Summary

This document covers a range of calculus topics, primarily focusing on differentiation, integration, limits, and applications of calculus.

1. Differentiation

• Optimization / Maximum Value: (Q38) Finding the maximum value of a function `f(x)` by setting its first derivative `f'(x)` to zero.
  • Example: For `f(x) = x³ - 27x + 15`, find `x` where `f'(x) = 0`.
• Derivatives of Parametric Equations: (Q39) Calculating `dy/dx` from `x = a sin t` and `y = a cos t` using the chain rule `(dy/dt) / (dx/dt)`.
  • Example: If `x = a sin t`, `y = a cos t`, then `y' = -tan t`.
• Derivative of Inverse Trigonometric Functions: (Q4) Applying the formula `d/dx (sec⁻¹ u) = 1 / (|u|√(u² - 1)) * du/dx`.
  • Example: `d/dx (sec⁻¹ (5x)) = 1 / (x√(25x² - 1))` (assuming `x > 0`).
• Higher-Order Derivatives (Product Rule & Chain Rule): (Q15) Calculating the second derivative `f''(x)` using repeated application of the product and chain rules.
  • Example: For `f(x) = xe⁻ˣ`, `f''(x) = (x - 2)e⁻ˣ`.
• Kinematics (Velocity from Position): (Q50) Determining velocity `v(t)` by differentiating the position function `x(t)` with respect to time, `v(t) = dx/dt`.
  • Example: If `x = 5t² - 2t³`, then `v(t) = 10t - 6t²`.
• Optimization (Maximizing Product): (Q16) Finding two positive numbers whose sum is constant and product is maximum, often solved by setting the derivative of the product function to zero.
  • Example: Sum = 105, numbers are 52.5 and 52.5. If restricted to integers, select options closest to the optimal value.

2. Integration

2.1 Indefinite Integrals

• Basic Trigonometric Integrals:
  • (Q40) `∫ sec x tan x dx = sec x + C`.
  • (Q48) `∫ sin x cos x dx = (sin² x)/2 + C` (using u-substitution).
  • (Q11) `∫ sec x dx = ln|sec x + tan x| + C`.
• Trigonometric Identities for Integration: (Q42) Using identities like `sin² x = (1 - cos 2x) / 2` to simplify integrals.
  • Example: `∫ sin² x dx = (1/2)x - (1/4)sin 2x + C`.
• Integration by Substitution (u-substitution):
  • (Q41) `∫ dx / (x - x ln x)`: Substitute `u = 1 - ln x`. Result: `-ln|1 - ln x| + C`.
  • (Q7) `∫ eˣ / (4 + eˣ) dx`: Substitute `u = 4 + eˣ`. Result: `ln(4 + eˣ) + C`.
  • (Q2) `∫ sin 2x / sin x dx`: Use `sin 2x = 2 sin x cos x`. Result: `2 sin x + C`.
• Integration by Parts: (Q43, Q13) Applying the formula `∫ u dv = uv - ∫ v du`.
  • (Q43) `∫ xeˣ dx = (x - 1)eˣ + C`.
  • (Q13) Requires careful application, particularly for `∫ x²eˣ dx` (used twice). *Note: The problem statement or options for Q13 appear inconsistent.*
• Reduction Formulas: (Q46) Using a given reduction formula to evaluate integrals of powers of trigonometric functions.
  • Example: For `I_n = ∫ cotⁿ x dx`, `I_n = -cotⁿ⁻¹ x / (n-1) - I_n-₂`.
• Partial Fractions: (Q18) Decomposing rational functions into simpler fractions before integrating.
  • Example: `∫ (x+1) / (x² - x - 6) dx = (1/5) ln[ (x-3)⁴ (x+2) ] + C`.
• Power Rule for Integration: (Q49) `∫ (ax+b)ⁿ dx` or `∫ xⁿ dx`. *Note: Q49 lacks definite integral limits for the numerical options.*

2.2 Definite Integrals

• Wallis' Formula: (Q5, Q6, Q12, Q17) Used for evaluating definite integrals of powers of sine and cosine functions over `[0, π/2]`.
  • Formula: `∫[0 to π/2] sinᵐ x cosⁿ x dx = [(m-1)!! (n-1)!!] / [(m+n)!!] * K`
  • `K = π/2` if both `m` and `n` are even; `K = 1` otherwise.
  • (Q5) `∫[0 to π/2] cos⁵ x sin² x dx = 8/105`.
  • (Q6) `∫[0 to π/2] sin⁵ x dx = 8/15`.
  • (Q12) `∫[0 to π/2] cos⁴ x dx = 3π/16`.
  • (Q17) `∫[0 to π/2] sin³ t dt = 2/3`. *Note: The correct answer 2/3 is not among the options for Q17.*
• Definite Integrals with Substitution and Parameters:
  • (Q3, Q9, Q10) Evaluating definite integrals involving trigonometric functions and finding an unknown parameter (`k` or `θ`). Requires careful application of u-substitution and evaluation of limits. *Note: Q3 appears to have a discrepancy in the problem statement or solution options.*
  • (Q14) `∫[0 to π/4] sec² x e^(4 tan x) dx`: Uses u-substitution `u = 4 tan x`. Result: `(1/4)(e⁴ - 1)`.
  • (Q20) `∫[-1 to 0] x√(x+1) dx`: Uses u-substitution `u = x+1`. Result: `-4/15`.
• Definite Integrals of Power Functions: (Q8) Evaluating definite integrals of `xⁿ` and `1/x`. *Note: The problem statement or options for Q8 appear inconsistent, as the direct calculation yields a result with `ln` terms, not an integer.*

3. Limits

• Standard Limits: (Q45) Evaluating known limits, such as `lim (x→0) (sin x / x)`.
  • Example: `lim (x→0) (sin x / x) = 1`.

4. Applications of Integration

• Area Between Curve and X-axis: (Q47) Finding the total area by integrating the absolute value of the function over intervals determined by its roots.
  • Example: For `y = x³ - 6x² + 8x`, roots at 0, 2, 4. Calculate `|∫[0 to 2] f(x) dx| + |∫[2 to 4] f(x) dx|`. Result: 8.

Summary of Common Discrepancies/Typographical Issues Noted:

• Q3, Q8, Q13, Q17: Calculation results do not match given integer or simplified options, suggesting potential typos in the problem statement, limits, or options provided.
• Q49: Lacks definite integral limits for a numerical answer.
• Q14: The integrand format `sec^2 xe^(4tmx)dx` likely implies `sec²(x) * e^(4 tan x) dx` based on the correct option.