Calculus

Topics

Limits, graphically & numerically — one-sided and two-sided limits, and when a limit fails to exist.

Evaluating limits — the limit laws, algebraic techniques, the Squeeze Theorem, and limits at infinity.

Continuity — the three-part definition, classifying discontinuities, and the Intermediate Value Theorem.

You'll be able to…

• Estimate a limit from a graph or a table of values.
• Evaluate limits with the limit laws and resolve \(\tfrac{0}{0}\) forms by factoring or rationalizing.
• Test continuity at a point with the three-part definition and classify any break.
• Find horizontal asymptotes by evaluating limits at \(\pm\infty\).

Worked example

Topics

The derivative as a limit — \(f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\), the tangent-line slope, and differentiability vs. continuity.

Differentiation rules — power, product, quotient, and chain rules; derivatives of trig, exponential, and logarithmic functions.

Implicit & inverse differentiation — \(\tfrac{dy}{dx}\) when \(y\) is tangled with \(x\), and derivatives of inverse functions.

You'll be able to…

• Compute a derivative from the limit definition and read it as a tangent slope.
• Differentiate fluently with the power, product, quotient, and chain rules.
• Differentiate \(\sin x\), \(\cos x\), \(e^x\), and \(\ln x\) from memory.
• Use implicit differentiation to find \(\tfrac{dy}{dx}\) for an implicit relation.

Worked example

Topics

Related rates & motion — position, velocity, and acceleration; rates linked through a shared equation.

Curve analysis — the first and second derivative tests, concavity, inflection points, and the Mean Value Theorem.

Optimization & approximation — absolute extrema in context, plus L'Hôpital's Rule for indeterminate forms.

You'll be able to…

• Find critical points and classify extrema with the first and second derivative tests.
• Determine intervals of increase, decrease, and concavity to sketch a curve.
• Set up and solve related-rates and optimization problems in context.
• Apply L'Hôpital's Rule to \(\tfrac{0}{0}\) and \(\tfrac{\infty}{\infty}\) limits.

Worked example

Topics

The definite integral — Riemann sums as a limit, antiderivatives, and both parts of the Fundamental Theorem of Calculus.

Techniques — \(u\)-substitution (and, for BC, integration by parts and partial fractions).

Applications — area between curves, volumes by disks, washers, and shells, and average value.

You'll be able to…

• Express a definite integral as a limit of Riemann sums.
• Apply both parts of the Fundamental Theorem to evaluate and differentiate integrals.
• Integrate by \(u\)-substitution and reverse the chain rule.
• Compute area between curves and a solid's volume by disks, washers, or shells.

Worked example

Topics

Differential equations — slope fields, separation of variables, and exponential growth and decay (AB & BC).

Series & convergence (BC) — the convergence tests, power series, and Taylor and Maclaurin series.

Parametric, polar & vector calculus (BC) — calculus on curves defined by \(x(t)\), \(y(t)\), and \(r(\theta)\).

You'll be able to…

• Solve a separable differential equation and sketch its slope field.
• Test a series for convergence with the appropriate test.
• Build the Maclaurin series for \(e^x\), \(\sin x\), and \(\cos x\).
• Differentiate and integrate parametric and polar curves (BC).

Worked example