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Calculus AP Final Review: Limits and Derivatives

Page 1: Understanding Limits

• Use the graph to determine values and limits:

  • a. f(-2) = 3.3

  • b. f(0) = undetermined

  • c. f(2) not given

  • d. lim f(x) as x approaches -3 = 3

  • e. lim f(x) as x approaches 2 = does not exist (dne)

  • f. lim f(x) as x approaches -1 from the right = 2

  • g. lim f(x) = not specified

  • h. lim f(x) = does not exist (DD)

  • i. lim f(x) as x approaches -1 from the left = d.n.e

  • j. lim f(x) = -2

  • k. lim f(x) as x approaches 2 from the right = unspecified

  • l. Discontinuities:

    • x = -3 (removable)

    • x = 0 (removable)

    • x = 2 (non-removable)

  • m. Reason for limit's non-existence at x = 2:

    • lim f(x) from the left does not equal lim f(x) from the right.

• Evaluate limits:

  1. lim (x^2+x-6)/(x-2) = 5

  2. lim (2-x)/(x-5) as x approaches 4 = 1

  3. DNE in some cases

Page 2: Continuity and Asymptotes

• Piecewise function continuity:

  • Given function, find value of k that makes f(x) continuous: e.g.

    • k = -2 (choose from options available)

• Identifying vertical asymptotes:

  • For f(x) = (x^2 + 3x - 4)/(x^2 + 3x - 4)

  • Factors: (x+4)(x-1); vertical asymptotes at x = -2

• Find various limits:

  1. lim (sin x)/x as x approaches 0 = 1

  2. lim (6sinx)/(x-2) and more as specified.

Page 3: Differentiation Techniques

• Limit definition of derivative:

  • f'(x) = lim [f(x+h) - f(x)]/h

  • For f(x) = x^2 + 2x, compute directly using substitutions.

• Derivative at a point:

  • Using specific values (e.g. at c = 5) find f'(5)

Page 4: Further Differentiation Practice

• Advanced derivatives:

  • Practice finding derivatives using product/quotient rules and second derivatives:

    • e.g. for f(x) = (2x + 3)^4, find f"(x) by applying the chain rule.

Page 5: Application of Derivatives

• Given f'(g(x))g'(x) specific evaluations

• Equations of tangent lines:

  • Identify slopes at given points and write equations of tangent lines.

Page 6: Applications of Derivatives and Asymptotes

• Maximizing area with constraints:

• Solve optimization problems with values of x, y using given constraints (like fence problem).

Page 7: Continuity and Extrema

• Critical points and their classification:

  • Identify local extrema, checking when f'(x)=0.

  • Points of inflection, the conditions under which they occur.

Page 8: Graph Behavior and Manufacturer Profit

• Concavity and profit maximization:

• Determine intervals of increase/decrease for functions and associated tangent line equations.