AP Calculus BC

Review of AP Calculus AB Core: Limits, Derivatives, and Integrals

Calculus BC

✓By the end of this lesson students will be able to recall and apply fundamental concepts of limits, continuity, and asymptotes.
✓By the end of this lesson students will be able to accurately compute derivatives using various rules and apply them to analyze function behavior and solve related rates and optimization problems.
✓By the end of this lesson students will be able to evaluate definite and indefinite integrals, including using U-substitution, and apply integration techniques to calculate areas and volumes.
✓By the end of this lesson students will be able to state and apply the Fundamental Theorem of Calculus and other key theorems like the Mean Value Theorem for both derivatives and integrals.
✓By the end of this lesson students will be able to identify and correct common misconceptions related to AB Calculus topics.

Key concepts

Limits and Continuity

A limit describes the behavior of a function as its input approaches a certain value. For a limit to exist, the left-hand and right-hand limits must be equal. Continuity at a point 'c' requires that the limit as x approaches 'c' exists, the function is defined at 'c', and these two values are equal. Discontinuities can be removable (hole) or non-removable (jump, infinite). L'Hopital's Rule can be used to evaluate indeterminate forms (0/0 or ∞/∞).

lim (x→c) f(x) = L; Continuity: lim (x→c) f(x) = f(c)

Derivatives: Definition and Rules

The derivative of a function f(x) represents the instantaneous rate of change of f(x) with respect to x, or the slope of the tangent line to the graph of f(x) at a given point. It is defined as a limit. Key rules include the Power Rule, Product Rule, Quotient Rule, and Chain Rule. Implicit differentiation is used for equations where y is not explicitly defined as a function of x.

f'(x) = lim (h→0) [f(x+h) - f(x)] / h

Applications of Derivatives

Derivatives are used to determine intervals where a function is increasing or decreasing (first derivative test), concavity (second derivative test), and local/absolute extrema. Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another. Optimization problems use derivatives to find maximum or minimum values of a function.

Integrals: Antiderivatives and Definite Integrals

An antiderivative F(x) of a function f(x) is a function whose derivative is f(x). Indefinite integrals represent the family of all antiderivatives, denoted by ∫f(x)dx = F(x) + C. A definite integral ∫(a to b) f(x)dx represents the net signed area between the function's graph and the x-axis from x=a to x=b. U-substitution is a technique for integrating composite functions.

∫f(x)dx = F(x) + C; ∫(a to b) f(x)dx = F(b) - F(a)

Fundamental Theorem of Calculus (FTC)

Part 1 of the FTC states that if F(x) is defined as the integral of f(t) from a to x, then F'(x) = f(x). Part 2 of the FTC provides a method for evaluating definite integrals: if F is any antiderivative of f, then the definite integral of f from a to b is F(b) - F(a).

Part 1: d/dx [∫(a to x) f(t)dt] = f(x); Part 2: ∫(a to b) f(x)dx = F(b) - F(a)

Applications of Integrals

Integrals are used to find the area between curves, the volume of solids of revolution using the disk and washer methods, and the volume of solids with known cross-sections. The average value of a function over an interval can also be found using integration.

Average Value = (1/(b-a)) ∫(a to b) f(x)dx

Key facts to remember

1The derivative f'(x) gives the slope of the tangent line and instantaneous rate of change.
2The integral ∫f(x)dx represents the antiderivative, and ∫(a to b) f(x)dx represents the net signed area.
3The Fundamental Theorem of Calculus (FTC) connects differentiation and integration: ∫(a to b) f(x)dx = F(b) - F(a) where F'(x) = f(x).
4L'Hopital's Rule applies to indeterminate forms 0/0 or ∞/∞ for limits.
5The Chain Rule is crucial for differentiating composite functions: d/dx [f(g(x))] = f'(g(x)) * g'(x).
6The Mean Value Theorem for Derivatives states that if f is continuous on [a,b] and differentiable on (a,b), there exists a c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a).
7The Mean Value Theorem for Integrals states that if f is continuous on [a,b], there exists a c in [a,b] such that f(c) = (1/(b-a)) ∫(a to b) f(x)dx.
8U-substitution is the reverse of the Chain Rule for integration.

Worked examples

Example 1

Evaluate the limit: lim (x→0) (e^(2x) - 1) / sin(x)

IFirst, check for indeterminate form: As x→0, e^(2x) - 1 → e^0 - 1 = 1 - 1 = 0. As x→0, sin(x) → sin(0) = 0. This is an indeterminate form 0/0, so L'Hopital's Rule can be applied.

IIApply L'Hopital's Rule: Take the derivative of the numerator and the derivative of the denominator.

IIId/dx (e^(2x) - 1) = 2e^(2x)

IVd/dx (sin(x)) = cos(x)

VNow, evaluate the limit of the new expression: lim (x→0) (2e^(2x)) / cos(x)

VISubstitute x=0: (2e^(2*0)) / cos(0) = (2e^0) / 1 = (2*1) / 1 = 2

Answer

L'Hopital's Rule is a powerful tool for indeterminate forms, but always check the form first.

Example 2

Find dy/dx for the equation x^2y + xy^2 = 6 using implicit differentiation.

IDifferentiate both sides of the equation with respect to x.

IId/dx (x^2y) + d/dx (xy^2) = d/dx (6)

IIIApply the Product Rule for d/dx (x^2y): (d/dx(x^2))y + x^2(d/dx(y)) = 2xy + x^2(dy/dx)

IVApply the Product Rule for d/dx (xy^2): (d/dx(x))y^2 + x(d/dx(y^2)) = 1y^2 + x(2y(dy/dx)) = y^2 + 2xy(dy/dx)

VThe derivative of the constant 6 is 0.

VICombine the differentiated terms: 2xy + x^2(dy/dx) + y^2 + 2xy(dy/dx) = 0

VIIGroup terms containing dy/dx: x^2(dy/dx) + 2xy(dy/dx) = -2xy - y^2

VIIIFactor out dy/dx: (dy/dx)(x^2 + 2xy) = -2xy - y^2

9Solve for dy/dx: dy/dx = (-2xy - y^2) / (x^2 + 2xy)

Answer

dy/dx = (-2xy - y^2) / (x^2 + 2xy)

Remember to apply the Chain Rule when differentiating terms involving 'y' with respect to 'x', resulting in a dy/dx term.

Example 3

Evaluate the definite integral: ∫(0 to 1) x * e^(x^2) dx

IIdentify a suitable U-substitution. Let u = x^2.

IIFind du/dx: du/dx = 2x, so du = 2x dx.

IIIAdjust the integral to fit 'du'. We have x dx, but need 2x dx. So, (1/2)du = x dx.

IVChange the limits of integration according to u = x^2:

VWhen x = 0, u = 0^2 = 0.

VIWhen x = 1, u = 1^2 = 1.

VIISubstitute u and du into the integral: ∫(0 to 1) (1/2)e^u du.

VIIIMove the constant (1/2) outside the integral: (1/2) ∫(0 to 1) e^u du.

9Integrate e^u: The antiderivative of e^u is e^u.

10Evaluate the definite integral using the new limits: (1/2) [e^u] from 0 to 1.

11Apply the Fundamental Theorem of Calculus: (1/2) (e^1 - e^0).

12Simplify: (1/2) (e - 1).

Answer

(e - 1) / 2

When performing U-substitution for definite integrals, it's often easiest to change the limits of integration to be in terms of 'u' to avoid substituting back 'x' later.

Common mistakes

✗Incorrectly applying L'Hopital's Rule when the limit is not an indeterminate form.
✗Forgetting to apply the Chain Rule, especially in implicit differentiation or when differentiating composite functions.
✗Not changing the limits of integration when performing U-substitution in definite integrals, or forgetting to substitute back 'x' if not changing limits.
✗Confusing the conditions for continuity versus differentiability (differentiability implies continuity, but not vice-versa).
✗Errors in algebraic manipulation, particularly when solving for dy/dx or simplifying integral expressions.
✗Misinterpreting the meaning of the definite integral (e.g., confusing net signed area with total area).

Exam tips

★Always show all steps in your work, especially for free-response questions, to earn partial credit.
★For limits, always attempt direct substitution first. If it yields an indeterminate form, then consider algebraic manipulation or L'Hopital's Rule.
★Memorize all differentiation and integration rules, including those for trigonometric, exponential, and logarithmic functions.
★Practice identifying the correct integration technique (e.g., U-substitution) and setting up integral applications (area, volume).
★Pay close attention to notation. Use correct mathematical symbols and clearly label your work.

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