Lesson: Introduction to Calculus

Duration: 50 minutes

Section 1: Limits and Continuity

Exposition: Calculus is a branch of mathematics that deals with change and motion. It is divided into two main branches: differential calculus and integral calculus. In this lesson, we will start by exploring the concept of limits and continuity, which are fundamental to understanding calculus.

Activity: 1. Let’s start by understanding the concept of a limit. Consider the function f(x) = 2x + 3. We want to find the limit of f(x) as x approaches 2. To do this, we can create a table of values by plugging in values of x that are getting closer and closer to 2. Let’s complete the table together:

x | f(x)

1.9 | 6.8 1.99 | 6.98 1.999| 6.998 2.001| 7.002 2.01 | 7.02 2.1 | 7.2

As x gets closer to 2, we can observe that f(x) gets closer to 7. This suggests that the limit of f(x) as x approaches 2 is 7.

1. Now, let’s discuss continuity. A function is said to be continuous if there are no breaks, jumps, or holes in its graph. For example, the function f(x) = x^2 is continuous everywhere. On the other hand, the function g(x) = 1/x is not continuous at x = 0 because it has a vertical asymptote.

Questions: 1. Find the limit of f(x) = 3x - 2 as x approaches 4. 2. Determine if the function h(x) = 2x + 1 is continuous at x = 5. 3. Find the limit of g(x) = (x^2 - 4)/(x - 2) as x approaches 2. 4. Is the function f(x) = |x| continuous at x = 0? 5. Find the limit of h(x) = (x^3 - 8)/(x - 2) as x approaches 2.

Section 2: Differentiation

Exposition: Differentiation is a process used to find the rate at which a function changes. It allows us to calculate the slope of a curve at any given point. The derivative of a function represents this slope and is denoted by f’(x) or dy/dx.

Activity: 1. Let’s start by finding the derivative of a simple function. Consider the function f(x) = 3x^2 + 2x. To find its derivative, we need to apply the power rule. The power rule states that if f(x) = x^n, then f’(x) = nx^(n-1). Let’s find the derivative of f(x) together:

f(x) = 3x^2 + 2x f’(x) = 2(3)x^(2-1) + 1(2)x^(1-1) = 6x + 2

1. Now, let’s discuss the concept of the chain rule. The chain rule is used when we have a composition of functions. For example, consider the function f(x) = (2x^2 + 3x)^3. To find its derivative, we need to apply the chain rule. The chain rule states that if f(x) = g(h(x)), then f’(x) = g’(h(x)) * h’(x). Let’s find the derivative of f(x) together:

   f(x) = (2x^2 + 3x)^3 f’(x) = 3(2x^2 + 3x)^2 * (4x + 3)

Questions: 1. Find the derivative of f(x) = 4x^3 - 2x^2 + 5x - 1. 2. Use the chain rule to find the derivative of f(x) = (3x^2 + 2x)^4. 3. Find the derivative of g(x) = √(2x + 1). 4. Use the power rule to find the derivative of f(x) = 5x^4 - 3x^2 + 2. 5. Find the derivative of h(x) = e^(2x).

Section 3: Integration

Exposition: Integration is the reverse process of differentiation. It allows us to find the area under a curve or the accumulation of quantities over a given interval. The integral of a function is denoted by ∫f(x) dx.

Activity: 1. Let’s start by finding the integral of a simple function. Consider the function f(x) = 2x + 3. To find its integral, we need to apply the power rule for integration. The power rule for integration states that if f(x) = x^n, then ∫f(x) dx = (1/(n+1))x^(n+1) + C, where C is the constant of integration. Let’s find the integral of f(x) together:

f(x) = 2x + 3 ∫f(x) dx = (1/2)x^2 + 3x + C

1. Now, let’s discuss the concept of definite integration. Definite integration allows us to find the exact area under a curve between two given points. For example, consider the function f(x) = x^2. To find the area under the curve between x = 1 and x = 3, we need to evaluate the definite integral ∫[1,3] f(x) dx. Let’s find the definite integral of f(x) together:

   f(x) = x^2 ∫[1,3] f(x) dx = [(1/3)x^3] [1,3] = (1/3)(3^3) - (1/3)(1^3) = 9 - (1/3) = 8 2/3

Questions: 1. Find the integral of f(x) = 4x^3 - 2x^2 + 5x - 1. 2. Evaluate the definite integral ∫[0,2] (3x^2 + 2x) dx. 3. Find the integral of g(x) = 1/x. 4. Evaluate the definite integral ∫[1,4] (2x + 3) dx. 5. Find the integral of h(x) = e^(2x).

Note: Remember to provide explanations and guide the student through each activity and question.