Whizmath Calculus Masterclass: The Mathematics of Change

Welcome to Whizmath's Comprehensive Calculus Lesson! Calculus is the mathematical study of continuous change, with two main branches: differential calculus (concerning rates of change and slopes) and integral calculus (concerning accumulation and areas).

From physics to economics, calculus helps us model and understand dynamic systems. This guide will take you from fundamental concepts to advanced applications.

Lesson Objectives

By the end of this lesson, you will:

Understand limits and continuity
Master differentiation rules and techniques
Learn integration methods and applications
Explore the Fundamental Theorem of Calculus
Solve optimization and related rates problems
Apply calculus to real-world scenarios

Section 1: Limits and Continuity

1.1 Understanding Limits

The limit describes the behavior of a function as the input approaches some value:

lim_x→a f(x) = L

Limit Properties

Sum Rule

lim(f(x) + g(x)) = lim f(x) + lim g(x)

Product Rule

lim(f(x) × g(x)) = lim f(x) × lim g(x)

Quotient Rule

lim(f(x)/g(x)) = lim f(x) / lim g(x) (if lim g(x) ≠ 0)

Example

Evaluate lim_x→2 (x² + 3x - 2)

Direct substitution: 2² + 3(2) - 2 = 4 + 6 - 2 = 8

1.2 Continuity

A function f is continuous at x = a if:

f(a) exists
lim_x→a f(x) exists
lim_x→a f(x) = f(a)

Continuous function (left) vs discontinuous function (right)

Section 2: Differential Calculus

2.1 Derivatives and Differentiation Rules

The derivative represents the rate of change of a function:

f'(x) = lim_h→0 [f(x+h) - f(x)]/h

Basic Differentiation Rules

Function	Derivative
c (constant)	0
xⁿ	nx^n-1
sin(x)	cos(x)
cos(x)	-sin(x)
e^x	e^x
ln(x)	1/x

Chain Rule

For composite functions: (f(g(x)))' = f'(g(x)) × g'(x)

Example

Find the derivative of f(x) = (3x² + 2)⁵

Using chain rule: 5(3x² + 2)⁴ × 6x = 30x(3x² + 2)⁴

2.2 Applications of Derivatives

Tangent Lines

Slope at point x = a is f'(a)

Optimization

Find maxima/minima by solving f'(x) = 0

Related Rates

Relate rates of change using derivatives

Section 3: Integral Calculus

3.1 Antiderivatives and Indefinite Integrals

The antiderivative F of a function f satisfies F' = f:

∫f(x)dx = F(x) + C

Basic Integration Rules

Function	Integral
xⁿ (n ≠ -1)	(xⁿ⁺¹)/(n+1) + C
1/x	ln\|x\| + C
e^x	e^x + C
sin(x)	-cos(x) + C
cos(x)	sin(x) + C

Fundamental Theorem of Calculus

Part 1: If F(x) = ∫_a^x f(t)dt, then F'(x) = f(x)

Part 2: ∫_a^b f(x)dx = F(b) - F(a)

3.2 Applications of Integration

Area Under Curve

A = ∫_a^b f(x)dx

Volume of Revolution

V = π∫_a^b [f(x)]²dx

Average Value

f_avg = (1/(b-a))∫_a^b f(x)dx

Section 4: Practice Problems

Beginner Level

1. Find the derivative of f(x) = 4x³ - 2x² + 5x - 7

2. Evaluate ∫(3x² + 2x - 1)dx

Intermediate Level

3. Use the chain rule to find the derivative of f(x) = sin(3x²)

4. Calculate the area under f(x) = x² between x = 0 and x = 2

Advanced Level

5. A spherical balloon is being inflated. Find the rate of increase of the surface area when the radius is 5cm, if the radius is increasing at 0.2cm/s.

6. Find the volume of the solid formed by rotating y = √x about the x-axis from x = 0 to x = 4

Conclusion

Calculus provides powerful tools for analyzing change and accumulation, with applications across science, engineering, economics, and beyond. By mastering these concepts, you'll develop a deeper understanding of dynamic systems and problem-solving techniques.

Continue your calculus journey with Whizmath! 🚀

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