Calculus 1 Final Exam Review

Topics Covered:

• Continuity
• Limits
• Derivatives
• Integration

Key Concepts and Problems

1. Evaluating Limits

• Direct Substitution: Check if substituting the value into the limit works directly.
• Factoring: Used when direct substitution leads to an indeterminate form (e.g., 0 in the denominator).
• Example:
  • Given limit problem lim_(x→3) [(x^2 - 9)/(x - 3)]:
  • Factor: (x^2 - 9) = (x + 3)(x - 3).
  • Cancel (x - 3), resulting in lim_(x→3) [(x + 3)].
  • Use substitution: 3 + 5 / 3 + 3 = 8 / 6 = 4 / 3.

2. Differentiation using Power Rule

• Power Rule: d/dx [x^n] = nx^(n-1).
• Example:
  • Differentiate x^4: Result is 4x^3.
  • Differentiate x^(-1): Result is -x^(-2) reversed for reciprocal.
  • Combine terms and bring exponents to the bottom where necessary.
• Final Example Result: 6x^5 - 3/x^2 + 1/(2√x).

3. Continuity in Piecewise Functions

• For a function f(x) to be continuous at x=c, both pieces of the function must be equal at x=c.
• Example:
  • Pieces: 2cx - 6 = x^2 + cx at x=3.
  • Solve: 2c(3) - 6 = 9 + 3c -> 6c - 6 = 9 + 3c -> 3c = 15 -> c = 5.

4. Derivative of Exponential and Logarithmic Functions

• Exponential: e^(u) derivative is e^(u) * u'.
• Logarithmic: ln(u) derivative is u'/u.
• Product Rule: (f*g)' = f'*g + f*g'.
• Example:
  • Differentiate e^(4x) * ln(2x+5):
  • Result: 4e^(4x)(ln(2x+5)) + e^(4x)*(2/(2x + 5)).

5. Basic Integration

• Power Rule for Integration: (x^n dx) = (x^(n+1))/(n+1) + C.
• Simplify expression before integrating.
• Example:
  • Simplify (4x^5 + x^4 - 3x^2)/x^2
  • Integrate each term: (4x^3 + x^2 - 3x).
• Combine: x^4/4 + (x^3/3) - (3x^2/2) + C.

6. Equation of Tangent Line

• Implicit Differentiation: Differentiate both sides with respect to x and solve for dy/dx.
• Use point (x, y) and slope (dy/dx) to form equation y - y1 = m(x - x1), where m is slope.
• Example: Implicit form differential 2x^3 + 4xy^2 + y^3 = 107 at (2,3).
• Find slope: Derivative calculation and plug in point.
• Solve equation: 16x + 25y = 107.

7. Limit Identities

• Definition of Derivative: f'(x) = lim_(h→0) [f(x+h) - f(x)] / h.
• Recognize limits involving derivatives and apply result directly.
• Example: Recognize such limit as derivative of sine (sin x), resulting in cos x.

8. U-Substitution in Integration

• U-Substitution: Make substitution to simplify integral; integrate, then substitute back.
• Example:
  • Integral of 2x√(3x^2 + 5) dx.
  • Let u = 3x^2 + 5, then, du = 6x dx or dx = du/(6x).
  • Result: 2/3(u^(3/2)) + C.

9. Related Rates

• Related Rates Problems: Differentiate related quantities with respect to time.
• Use volume formulas for geometric shapes (e.g., cylinder V=πr^2h).
• Example:
  • dh/dt = 3 ft/min, r=3ft, find dV/dt.
  • Calculate based on differential equation: dV/dt = π * 9 * 3 = 27π ft³/min.

10. Increasing and Decreasing Intervals

• First Derivative Test: Find f'(x), set to zero for critical points, and test intervals.
• Positive derivative -> Function increasing; Negative derivative -> Function decreasing.
• Example:
  • Given critical points, evaluate derivative sign on intervals to find increasing/decreasing.

11. Locating Maximum/Minimum Values

• Second Derivative Test: Find f''(x), set to zero for inflection points.
• Positive f''(x) -> Concave up; Negative f''(x) -> Concave down.
• Example: Parabola -x²+16x+5 has max point found by solving f'(x)=0.
• Maximum at (8, 69).

12. Average Value of a Function

• Formula: Average value = 1/(b-a) ∫_[a to b] f(x) dx.
• Use integration over interval and divide by interval length.
• Example: f(x) = x^3 + 8x - 4 from [1, 5].
• Result: 59 for average value.

13. Chain Rule for Composite Functions

• Chain Rule: d/dx f(g(x)) = f'(g(x)) * g'(x).
• Example: Differentiate (2x^3 - 7x^2)^8.
  • Result: 8(2x^3 - 7x^2)^7 * (6x^2 - 14x) simplified.

14. Evaluating Limits

• Simplify Complex Fractions: Clear fractions by multiplying by common denominator.
• Example: limit involving fractions simplifies to cancellation method.
  • Result: Evaluate remaining limit function directly.

15. Concavity and Inflection Points

• Second Derivative: f''(x) > 0 (concave up), f''(x) < 0 (concave down).
• Example: Determine concavity over interval.
  • Resulting intervals based on second derivative sign.