Limits & Continuity
Limit: \(\lim_{x\to a} f(x) = L\) if values of \(f(x)\) get arbitrarily close to \(L\) as \(x\) approaches \(a\). A function is continuous at \(a\) when \(\lim_{x\to a} f(x) = f(a)\).
Calculus — Learn by Doing
From limits and continuity to derivatives, integrals, and optimization. Use the interactive tools and examples below to build strong intuition and exam‑ready skills.
Limits
Continuity
Derivatives
Integrals
Optimization
Curve Sketching
Common Limits
| Form | Limit |
|---|---|
| \(\displaystyle\lim_{x\to 0} \frac{\sin x}{x}\) | 1 |
| \(\displaystyle\lim_{x\to 0} \frac{e^x-1}{x}\) | 1 |
| \(\displaystyle\lim_{x\to \infty} (1+\tfrac{1}{x})^x\) | e |
Derivatives: Rules & Calculators
Derivative of \(f(x)\) at \(a\): \(f'(a)=\lim_{h\to0} \tfrac{f(a+h)-f(a)}{h}\). Key rules: Power, Sum, Product, Quotient, and Chain.
Product Rule
\((fg)'=f'g+fg'\)
Quotient Rule
\(\left(\tfrac{f}{g}\right)'=\tfrac{f'g-fg'}{g^2}\)
Chain Rule
If \(y=f(g(x))\), then \(y'=f'(g(x))\cdot g'(x)\).
Common Derivatives
| Function | Derivative |
|---|---|
| \(x^n\) | \(nx^{n-1}\) |
| \(e^x\) | \(e^x\) |
| \(\ln x\) | \(1/x\) |
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(-\sin x\) |
Integrals
Indefinite integral: antiderivative \(\int f(x)\,dx=F(x)+C\). Definite integral: area as a limit of sums: \(\int_a^b f(x)\,dx\). We compute \(definite\) integrals numerically here.
Graph: Function, Tangent & Inflection
Optimization & Curve Sketching
Critical points: solve \(f'(x)=0\). Max/min via first or second derivative tests. Inflection where concavity changes (\(f''\) sign change).
End‑of‑Unit Quiz: Derivatives & Integrals
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Logarithms: Why \(\log(0)\) is undefined
\(\log_b(x)=y\iff b^y=x\) (for \(b>0,b\neq1,x>0\)). There is no real \(y\) such that \(b^y=0\), and \(\lim_{x\to0^+}\log_b x=-\infty\).
Core Calculus Concepts — Expanded Explanations
These explanations deepen understanding of the most important ideas in calculus, linking concepts and giving clearer intuition.
Limits
Limits describe the value a function approaches as the input gets closer to a specific point. They let us analyze functions at points where they may be undefined or behave strangely. Limits form the basis of derivatives and integrals.
- Key idea: focus on the value the function approaches, not the value it equals.
- Example: As x approaches 0, sin(x)/x approaches 1.
- Used to define continuity and derivatives.
Continuity
A function is continuous at a point when the limit exists, the function value exists, and both match. Continuous functions have no breaks or jumps.
- Types of discontinuity include removable, jump, and infinite.
- Continuity is required for many important theorems.
Derivatives
The derivative measures how fast a function changes. It represents the slope of the tangent line and describes instantaneous rate of change.
- Interpretations: slope, velocity, growth rate, sensitivity.
- Rules include power, product, quotient, and chain rules.
- Example: the derivative of position gives velocity.
Integrals
Integrals measure accumulation, such as area under a curve or total change over an interval. Definite integrals give a number, while indefinite integrals give families of functions.
- Describe area, volume, probability, and physical quantities.
- Connected to derivatives through the Fundamental Theorem of Calculus.
- Example: the integral of 2x from 0 to 3 is 9.
Optimization
Optimization finds maximum and minimum values of functions. Calculus identifies critical points where the derivative is zero or undefined.
- Use first and second derivative tests.
- Important in economics, engineering, design, and physics.
- Example: maximizing area with fixed perimeter.
Product Rule
The product rule is used when differentiating two functions that are multiplied together. Since both may change, both rates of change matter.
- Formula: (f g)' = f' g + f g'.
- Useful when expressions have variable times variable.
Quotient Rule
The quotient rule differentiates the ratio of two changing functions. It accounts for how the numerator and denominator both change.
- Formula: (f/g)' = (f' g - f g') / g².
- Important when the denominator contains a variable.
Chain Rule
The chain rule differentiates composite functions, where one function is inside another. Change flows through layers.
- Formula: (f(g(x)))' = f'(g(x)) × g'(x).
- Common in powers of expressions, trig of polynomials, and exponentials.
Core Calculus Concepts — Diagrams, Applets, Mini-Quizzes
Brief explanations with interactive sliders and a quick check question.
Limits
Limits describe the value a function approaches near a point. Example function: f(x) = (x² − 1)/(x − 1) has a removable hole at x = 1.
1
Mini-quiz: For f(x) = (x²−1)/(x−1), what is the limit as x → 1?
Continuity
A function is continuous at a if the limit exists and equals the function value there. A removable hole can be fixed by defining f(a) to equal the limit.
Mini-quiz: If a function has a hole at x=a but the limit exists, can we make it continuous at a by defining f(a) to that limit?
Derivatives
Derivatives measure instantaneous rate of change (slope of the tangent). Example: f(x) = x³ − 3x² + 2.
1
Mini-quiz: For f(x) = x³ − 3x² + 2, what is f'(1)?
Integrals
Integrals measure accumulation (area under a curve). We visualize area under f(x)=x² from 0 to b.
2
Mini-quiz: What is the exact value of the integral from 0 to 1 of x² dx?
Optimization
Find maxima/minima by solving f'(x)=0 and checking second derivative or sign changes. Example quadratic: f(x)=a x² + b x + c.
Mini-quiz: If a > 0 for f(x)=a x² + b x + c, is the vertex a maximum or a minimum?
Rules of Differentiation
Product: (fg)' = f'g + f g'. Quotient: (f/g)' = (f'g - f g') / g². Chain: If y=f(g(x)), then y' = f'(g(x)) · g'(x).
Mini-quiz: Product rule — d/dx [ x · e^x ] = ?
Mini-quiz: Quotient rule — d/dx [ x² / (x+1) ] = ?
Mini-quiz: Chain rule — d/dx [ (2x+1)³ ] = ?