Calculus Mastery
Explore the mathematics of change and accumulation — limits, derivatives, integrals, and the powerful techniques that unify them.
Key Formulas & Concepts
- Power Rule: $$\frac{d}{dx}\left[x^n\right] = n\,x^{n-1}$$
- Chain Rule: $$\frac{d}{dx}\left[f(g(x))\right] = f'(g(x)) \cdot g'(x)$$
- Product Rule: $$\frac{d}{dx}\left[f \cdot g\right] = f'\,g + f\,g'$$
- Quotient Rule: $$\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'\,g - f\,g'}{g^2}$$
- Integration by Parts: $$\int u\,dv = uv - \int v\,du$$
- L'Hôpital's Rule: If \(\lim \frac{f}{g}\) is \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then $$\lim \frac{f}{g} = \lim \frac{f'}{g'}$$
- Fundamental Theorem of Calculus: $$\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where } F' = f$$
- Common Integrals: $$\int e^x\,dx = e^x + C, \quad \int \frac{1}{x}\,dx = \ln|x| + C, \quad \int \sin x\,dx = -\cos x + C$$
Evaluate: \(\displaystyle\lim_{x \to 0} \frac{\sin 3x}{\tan 5x}\).
Show Hint
Rewrite as \(\dfrac{\sin 3x}{3x} \cdot \dfrac{5x}{\tan 5x} \cdot \dfrac{3}{5}\). Each ratio approaches 1 as \(x \to 0\).
Find \(\dfrac{dy}{dx}\) if \(y = \ln(\sin(e^{2x}))\).
Show Hint
Apply the chain rule layer by layer: \(\frac{d}{dx}\ln(u) = \frac{u'}{u}\), then \(\frac{d}{dx}\sin(v) = \cos(v) \cdot v'\), then \(\frac{d}{dx}e^{2x} = 2e^{2x}\).
Evaluate: \(\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\).
Show Hint
Direct substitution gives \(\frac{0}{0}\). Apply L'Hôpital's once to get \(\frac{e^x - 1}{2x}\), still \(\frac{0}{0}\). Apply again.
Evaluate: \(\displaystyle\int x^2 e^x\,dx\).
Show Hint
Let \(u = x^2\), \(dv = e^x\,dx\). You'll need to apply integration by parts twice. Each time, the power of \(x\) decreases by 1.
A spherical balloon is being inflated at a rate of \(100\;\text{cm}^3/\text{s}\). How fast is the radius increasing when the radius is 5 cm? (\(V = \frac{4}{3}\pi r^3\))
Show Hint
Differentiate \(V = \frac{4}{3}\pi r^3\) with respect to time: \(\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt}\). Plug in \(\frac{dV}{dt} = 100\) and \(r = 5\).
Find the area enclosed between \(y = x^2\) and \(y = 2x + 3\).
Show Hint
First find intersection points by solving \(x^2 = 2x + 3 \Rightarrow x^2 - 2x - 3 = 0\). Then integrate \((2x + 3 - x^2)\) between the two roots.
If \(x^3 + y^3 = 6xy\), find \(\dfrac{dy}{dx}\) and the equation of the tangent line at \((3, 3)\).
Show Hint
Differentiate both sides: \(3x^2 + 3y^2\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}\). Solve for \(\frac{dy}{dx}\), then evaluate at \((3,3)\).
Determine whether \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\) converges. If so, find its value.
Show Hint
Evaluate \(\displaystyle\lim_{b \to \infty} \int_1^b x^{-2}\,dx = \lim_{b \to \infty} \left[-\frac{1}{x}\right]_1^b = \lim_{b \to \infty} \left(-\frac{1}{b} + 1\right)\).
Find the first four nonzero terms of the Maclaurin series for \(f(x) = e^x \sin x\).
Show Hint
Multiply the series for \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\) and \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\). Collect terms by power of \(x\).
A farmer wants to fence a rectangular area of \(1800\;\text{m}^2\) and divide it into three equal pens with fences parallel to one side. What dimensions minimize the total fencing?
Show Hint
Let \(x\) be the width and \(y\) the length. Then \(xy = 1800\) and total fencing \(= 2y + 4x\). Substitute \(y = \frac{1800}{x}\) and minimize.
Evaluate: \(\displaystyle\int \frac{dx}{x^2\sqrt{x^2 + 9}}\).
Show Hint
Let \(x = 3\tan\theta\), so \(dx = 3\sec^2\theta\,d\theta\) and \(\sqrt{x^2+9} = 3\sec\theta\). Simplify and integrate.
Solve the differential equation \(\dfrac{dy}{dx} + 2y = e^{-x}\) with initial condition \(y(0) = 1\).
Show Hint
This is a first-order linear ODE. The integrating factor is \(e^{\int 2\,dx} = e^{2x}\). Multiply both sides by \(e^{2x}\) and integrate. Apply the initial condition to find \(C\).