Calculus Final Exam Review

Topics Covered

• Continuity
• Limits
• Derivatives
• Integration

Key Concepts and Examples

1. Evaluating Limits

• Example Problem: Evaluate (\lim_{x \to 3} \frac{x^2 - 9}{x-3}).
  • Direct Substitution: Not possible if it results in division by zero.
  • Solution: Factor the expression -> ((x+3)(x-3)), cancel (x-3).
  • Result: (\frac{x+5}{x+3}) -> Direct substitution gives (\frac{8}{6} = \frac{4}{3})._

2. Derivatives Using Power Rule

• Power Rule: (d/dx[x^n] = nx^{n-1}).
• Example Problem: Differentiate (x^6 + \frac{3}{x} + \sqrt{x}).
  • Steps:
    • Apply power rule to each term.
    • Rewrite terms to use power rule: (3x^{-1} = 3x^{-1}), (\sqrt{x} = x^{1/2}).
    • Simplify derivative.

3. Continuity

• Piecewise Functions: (f(x) = 2cx - 6) and (x^2 + cx).
  • To Make Continuous: Set parts equal at the point of interest and solve for (c).
  • Example Solution: Substitute and solve resulting equations for (c).

4. Derivatives of Exponential and Logarithmic Functions

• Exponential Functions: (d/dx[e^u] = e^u u').
• Logarithmic Functions: (d/dx[\ln(u)] = \frac{u'}{u}).
• Product Rule: (d/dx[f , g] = f'g + fg').
  • Example Solution: Differentiate (e^{4x}\ln(2x + 5)) using rules.

5. Evaluating Integrals

• Simplifying Before Integrating: Divide numerator by denominator term-wise.
• Power Rule for Integrals: (\int x^n , dx = \frac{x^{n+1}}{n+1} + C).
  • Example Problem: (\int (4x^5 + x^4 - 3x^2)/x^2 , dx).
  • Solution: Simplify, apply power rule, include constant of integration.

6. Tangent Lines Using Implicit Differentiation

• Find Equation: Use point and slope from implicit differentiation.
• Example Problem: (x^3 + 4xy^2 + y^3 = 107) at (2, 3).
  • Steps: Differentiate implicitly, solve for derivative at point, use point-slope form.

7. Limits and Derivatives

• Recognizing Derivative Form: (f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}).
  • Example Recognition: Recognize and solve derivative form for given function._

8. Integral Using Substitution

• U-Substitution: Simplify complex integrals by substituting variables.
  • Example Problem: (\int , 2x \sqrt{3x^2 + 5} , dx).
  • Solution: Use (u = 3x^2 + 5), solve for integral in terms of (u).

9. Related Rates

• Application: Volume changes in a cylinder as water is added.
  • Example Problem: Cylinder with diameter 6 ft, height increasing at 3 ft/min.
  • Solution: Use volume formula, differentiate, substitute known rates.

10. Critical Points and Intervals

• Finding Critical Points: Set first derivative to zero.
• Increasing/Decreasing Intervals: Use sign of first derivative to determine.

11. Maximum Values

• Parabolic Functions: Analyze using vertex form and derivatives.
  • Example: (f(x) = 16x - x^2 + 5), find max value at critical point.

12. Average Value of Function

• Formula: (\frac{1}{b-a} \int_a^b f(x) , dx).
  • Example: Over interval [1, 5], calculate average of given function.

13. Using Chain Rule

• Chain Rule: Differentiate composite functions effectively.
  • Example: (\frac{d}{dx}[u^8] = 8u^7 \cdot u'), apply to composite functions.

14. Limits with Complex Fractions

• Simplifying: Multiply numerator and denominator by common factor to simplify.
  • Evaluate Limit: Factor, simplify, use direct substitution if possible.

15. Concavity and Inflection Points

• Second Derivative Test: Use sign of second derivative to determine concavity.
  • Inflection Points: Points where concavity changes.

Tips for Success

• Practice problems with different techniques regularly.
• Understand the foundational rules and when to apply each calculus rule.
• Use visualization when solving problems involving geometry or graphs.