Interactive Calculus

Welcome to this interactive guide to calculus, the mathematics of continuous change. This module is designed to make abstract concepts tangible by allowing you to directly engage with the fundamental principles. Through the hands-on tools, you will visually explore limits, derivatives, and integrals to build a strong intuition for how calculus works and lay a solid foundation for more advanced topics.

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Learning Objectives

  • Evaluate limits numerically, graphically, and algebraically.
  • Calculate derivatives using the limit definition and apply them to analyze a function's behavior (e.g., intervals of increase/decrease, concavity).
  • Approximate the area under a curve using Riemann sums and understand its connection to the definite integral.
  • Apply fundamental rules and techniques, including the product rule, chain rule, and u-substitution, to find derivatives and integrals.
  • Visually confirm the First Fundamental Theorem of Calculus by observing that the derivative of an integral function is the original function.

The Limit of a Function

A limit describes the value a function *approaches* as its input gets closer to a certain point. The limit can exist even if the function is undefined at that point. Choose a function, then use the slider to move the point 'x' closer to the point of interest 'a' and observe how f(x) behaves.

Left Limit: -
Limit: -
Right Limit: -

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Techniques for Computing Limits

While observing the graph gives an intuition for a limit, these algebraic techniques are used to find limits precisely.

Principle 1: Direct Substitution

Always try this first. If a function is continuous at the point of interest (i.e., no holes, jumps, or asymptotes), the limit is simply the function's value at that point.

Example: For $\lim_{x \to 2} (x^2 + 3)$, we can substitute $x=2$ to get $(2)^2 + 3 = 7$.

Principle 2: Handling Indeterminate Form ($\frac{0}{0}$)

When direct substitution results in $\frac{0}{0}$, it means there's a "hole" in the graph. We can use algebraic techniques to simplify the expression and cancel the term causing the zero in the denominator.

Technique A: Factoring

Example: For $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$, factor to $\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2}$. Cancel $(x-2)$ to get $\lim_{x \to 2} (x+2) = 4$.

Technique B: Rationalizing

Example: For $\lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1}$, multiply by the conjugate $\frac{\sqrt{x}+1}{\sqrt{x}+1}$ to get $\lim_{x \to 1} \frac{x-1}{(x-1)(\sqrt{x}+1)}$. Cancel $(x-1)$ to get $\lim_{x \to 1} \frac{1}{\sqrt{x}+1} = \frac{1}{2}$.

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The Derivative

The derivative measures the instantaneous rate of change (the slope of the tangent line), while the average rate of change is the slope of the secant line between two points. Use the slider to change the point of tangency and see how the tangent line's slope compares to the secant line's slope.

Avg. Rate of Change on [0.0, 2.5]:
Inst. Rate of Change at '$a$':

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Computing Derivatives with Limits

The formal definition of a derivative is a limit. Specifically, it's the limit of the average rate of change as the interval shrinks to zero.

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

Example: Find the derivative of $f(x) = x^2$

  1. Set up the limit:

    $$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$$

  2. Expand the expression:

    $$f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$$

  3. Simplify by canceling terms:

    $$f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}$$

  4. Factor out h from the numerator:

    $$f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}$$

  5. Cancel h and evaluate the limit:

    $$f'(x) = \lim_{h \to 0} (2x + h) = 2x$$

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Differentiation Rules

While the limit definition is fundamental, it's often inefficient for calculations. We use a set of rules to find derivatives more quickly. Here are some of the most common ones.

The Power Rule

The simplest rule for differentiating polynomials.

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Example: If $f(x) = x^4$, then $f'(x) = 4x^3$.

The Product Rule

Used to differentiate the product of two functions.

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

Example: If $h(x) = x^2\sin(x)$, then $h'(x) = x^2\cos(x) + 2x\sin(x)$.

The Quotient Rule

Used to differentiate the quotient of two functions.

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$$

Example: If $h(x) = \frac{x}{\sin(x)}$, then $h'(x) = \frac{\sin(x) - x\cos(x)}{\sin^2(x)}$.

The Chain Rule

Used to differentiate composite functions.

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

Example: If $h(x) = (x^2 + 1)^3$, then $h'(x) = 3(x^2+1)^2(2x) = 6x(x^2+1)^2$.

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Graphs

A function's first derivative ($f'$) tells us where it's increasing or decreasing, while the second derivative ($f''$) tells us about its concavity. Select a function to see its graph and the corresponding graphs and sign charts for its derivatives.

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The Definite Integral

The definite integral represents the area under a curve. We can approximate this area using a Riemann Sum, which adds up the areas of multiple shapes. Adjust the number of shapes ('$n$') and the approximation method to see how it affects the accuracy of the estimated area.

Approx. Area:
Shape # Interval Height Area
Total Approximate Area:

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Integration Techniques

Many integration techniques can be seen as differentiation rules in reverse. Understanding this connection can provide a deeper intuition for how they work.

Power Rule & Its Reverse

The simplest differentiation rule, the Power Rule, has a direct counterpart for integration. Notice how the steps for integration are the exact reverse of the steps for differentiation.

Differentiation: $\frac{d}{dx}(x^n)$
  1. Multiply by the exponent: $n \cdot x^n$
  2. Decrease the exponent by 1: $nx^{n-1}$
Integration: $\int x^n dx$
  1. Increase the exponent by 1: $x^{n+1}$
  2. Divide by the new exponent: $\frac{x^{n+1}}{n+1}$

Chain Rule & U-Substitution

The Chain Rule, $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, is used for nested functions. U-Substitution is its reverse, simplifying an integral that contains a function and its derivative by changing variables.

Demo: $\int 2x \cos(x^2) dx$

1. Let $u = x^2$
2. Then $du = 2x dx$
3. Substitute: $\int \cos(u) du$
4. Integrate: $\sin(u) + C$
5. Back-substitute: $\sin(x^2) + C$

Product Rule & Integration by Parts

Integration by Parts is derived from the Product Rule, $\frac{d}{dx}[uv] = u v' + v u'$. By integrating both sides, we get the formula $\int u \, dv = uv - \int v \, du$.

Demo: $\int x \cos(x) dx$

1. Let $u = x$ and $dv = \cos(x) dx$
2. Then $du = dx$ and $v = \sin(x)$
3. Apply $uv - \int v du$: $x$$\sin(x)$ - $\int$$\sin(x)$ $dx$
4. Integrate: $x\sin(x) - (-\cos(x)) + C$
5. Simplify: $x\sin(x) + \cos(x) + C$

Putting It All Together: Second FTC

Once you find the antiderivative $F(x)$ using a technique, the Second Fundamental Theorem of Calculus lets you evaluate the definite integral from $a$ to $b$.

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

Example: To compute $\int_0^{\pi/2} x \cos(x) dx$:

  1. Find the antiderivative using Integration by Parts: $F(x) = x\sin(x) + \cos(x)$.
  2. Evaluate at the upper limit: $F(\pi/2) = (\pi/2)\sin(\pi/2) + \cos(\pi/2) = \pi/2 \cdot 1 + 0 = \pi/2$.
  3. Evaluate at the lower limit: $F(0) = 0\cdot\sin(0) + \cos(0) = 0 + 1 = 1$.
  4. Subtract: $F(\pi/2) - F(0) = \pi/2 - 1 \approx 0.571$.

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Fundamental Theorem of Calculus

The First FTC establishes a powerful link between derivatives and integrals. It states that if a function $G(x)$ is defined as an "area accumulation" function, $G(x) = \int_a^x f(t) dt$, then the rate of change of this area function is the original function itself: $G'(x) = f(x)$.

Use the sliders to change the start and end points of the integral. Observe the two key relationships demonstrated below:

$G(x) = \int_a^x f(t) dt$

$f(x)$

Value of $G(x)$:
Area under $f(t)$:
Slope of $G(x)$ at $x$:
Value of $f(x)$:

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