Before You Begin

Key Concepts & Formulas to Remember

Limits & Continuity

A function is continuous at x = a if the limit exists, equals f(a), and f(a) is defined.

$$\lim_{x\to a}f(x) = L \iff \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L$$

Derivative Definition

The derivative is the instantaneous rate of change — slope of the tangent line.

$$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

Chain Rule

Differentiate composite functions: derivative of outer times derivative of inner.

$$\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)$$

Fundamental Theorem of Calculus

Links differentiation and integration — two core operations of calculus.

$$\frac{d}{dx}\int_a^{x}f(t)\,dt = f(x)$$$$\int_a^b f(x)\,dx = F(b)-F(a)$$

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), then there exists c such that:

$$f'(c) = \frac{f(b)-f(a)}{b-a}$$

Related Rates & Optimization

Use implicit differentiation with respect to time. Set f'(x) = 0 for critical points.

$$\frac{dA}{dt} = \frac{dA}{dx}\cdot\frac{dx}{dt}$$

Integration by Parts

Used when the integrand is a product of two functions. Choose u and dv strategically (LIATE).

$$\int u\,dv = uv - \int v\,du$$

Infinite Series & Convergence

A series $\sum a_n$ converges if partial sums approach a finite limit. Key tests: ratio, comparison, alternating.

$$\text{Ratio Test: } \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|= L$$

Taylor & Maclaurin Series

Power series representation of a function centered at x = a (Maclaurin: a = 0).

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$

Parametric & Polar Calculus

Slope of parametric curve; arc length in polar coordinates.

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\quad A=\tfrac{1}{2}\int r^2\,d\theta$$

Time Remaining

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Q 1 / 20

Score

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Limits · AB/BC

Medium

📌 L'Hôpital's Rule applies when a limit produces the indeterminate form 0/0 or ∞/∞.

What is $\displaystyle\lim_{x \to 0} \frac{\sin(3x)}{x}$?

Continuity · AB/BC

Medium

📌 For continuity at x = a: (1) f(a) exists, (2) limit exists, (3) they are equal.

Let $f(x) = \begin{cases} x^2 - 1 & x \neq 2 \\ 3 & x = 2 \end{cases}$. Which statement is true about $f$ at $x = 2$?

Derivatives · AB/BC

Medium

📌 Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x). Apply to composite functions.

If $f(x) = \sin(x^2 + 1)$, find $f'(x)$.

Implicit Differentiation · AB/BC

Hard

📌 Differentiate both sides with respect to x; treat y as a function of x and apply chain rule to y-terms.

Given $x^2 + y^2 = 25$, find $\dfrac{dy}{dx}$.

Mean Value Theorem · AB/BC

Medium

📌 MVT guarantees c ∈ (a,b) where the instantaneous rate equals the average rate of change.

Let $f(x) = x^3 - 3x$ on $[0, 2]$. What value of $c$ in $(0,2)$ satisfies the Mean Value Theorem?

Curve Analysis · AB/BC

Medium

📌 f is concave up where f″ > 0, concave down where f″ < 0. Inflection points: f″ changes sign.

For $f(x) = x^4 - 4x^3$, on which interval(s) is $f$ concave up?

Related Rates · AB/BC

Hard

📌 Identify all variables as functions of time t. Differentiate the geometric relationship implicitly with respect to t.

A spherical balloon is being inflated so that its volume increases at $100\ \text{cm}^3/\text{s}$. How fast is the radius increasing when the radius is $5\ \text{cm}$? (Volume of sphere: $V = \tfrac{4}{3}\pi r^3$)

Definite Integrals · AB/BC

Medium

📌 FTC Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f.

Evaluate $\displaystyle\int_{0}^{2} (3x^2 - 2x + 1)\,dx$.

FTC & Accumulation · AB/BC

Hard

📌 FTC Part 1: d/dx ∫ₐˣ f(t) dt = f(x). With chain rule: d/dx ∫ₐᵍ⁽ˣ⁾ f(t) dt = f(g(x))·g′(x).

Let $G(x) = \displaystyle\int_{1}^{x^2} \sqrt{t^3 + 1}\,dt$. Find $G'(x)$.

U-Substitution · AB/BC

Medium

📌 For ∫f(g(x))g′(x)dx, let u = g(x), du = g′(x)dx. Transform and integrate in u, then back-substitute.

Evaluate $\displaystyle\int x\cos(x^2)\,dx$.

Area Between Curves · AB/BC

Medium

📌 Area = ∫ₐᵇ [f(x) − g(x)] dx where f(x) ≥ g(x) on [a, b]. Find intersections first.

Find the area of the region bounded by $y = x^2$ and $y = x$.

Differential Equations · AB/BC

Hard

📌 Separate variables: move all y-terms to one side and all x-terms to the other, then integrate both sides.

Solve the differential equation $\dfrac{dy}{dx} = \dfrac{2x}{y}$ with initial condition $y(0) = 3$.

L'Hôpital's Rule · AB/BC

Medium

📌 If lim gives 0/0 or ∞/∞, apply L'Hôpital: lim f/g = lim f′/g′. May need to apply repeatedly.

Evaluate $\displaystyle\lim_{x\to 0}\frac{e^x - 1 - x}{x^2}$.

Volume of Revolution · AB/BC

Hard

📌 Disk method (about x-axis): V = π ∫ₐᵇ [f(x)]² dx. Shell method (about y-axis): V = 2π ∫ₐᵇ x·f(x) dx.

The region bounded by $y = \sqrt{x}$, $y=0$, and $x=4$ is revolved about the $x$-axis. Find the volume.

Integration by Parts · BC

Hard

📌 LIATE order for choosing u: Logarithmic → Inverse trig → Algebraic → Trig → Exponential.

Evaluate $\displaystyle\int x e^x\,dx$.

Series Convergence · BC

Hard

📌 Ratio Test: if lim|aₙ₊₁/aₙ| = L < 1, the series converges absolutely; L > 1 diverges; L = 1 inconclusive.

Does the series $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{2^n}$ converge or diverge?

Taylor Series · BC

Hard

📌 Maclaurin for eˣ = Σ xⁿ/n!. Substitute −x² to get the series for e^(−x²).

Which of the following is the Maclaurin series for $e^{-x^2}$?

Parametric Calculus · BC

Hard

📌 For parametric curves: dy/dx = (dy/dt)/(dx/dt). For arc length: L = ∫√[(dx/dt)²+(dy/dt)²] dt.

A curve is defined parametrically by $x = t^2$ and $y = t^3$. Find $\dfrac{dy}{dx}$ at $t = 2$.

Polar Area · BC

Hard

📌 Area enclosed by polar curve r = f(θ): A = ½∫αᵝ r² dθ. Find bounds from the curve's extent.

Find the area enclosed by the polar curve $r = 2\cos\theta$ for $0 \le \theta \le \pi$.

Improper Integrals · BC

Hard

📌 Improper integral: replace ∞ with a limit variable b and take lim as b→∞. Converges if limit is finite.

Evaluate $\displaystyle\int_{1}^{\infty} \frac{1}{x^2}\,dx$.

Key Concepts & Formulas to Remember

Limits & Continuity

Derivative Definition

Chain Rule

Fundamental Theorem of Calculus

Mean Value Theorem

Related Rates & Optimization

Integration by Parts

Infinite Series & Convergence

Taylor & Maclaurin Series

Parametric & Polar Calculus

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