1.5 Calculus
Created Date: 2025-05-10
Just quickly familiarize yourself with the core concepts of calculus and recall the math classes you have taken. If you have not studied it, you need to read the recommended textbooks in detail.
1.4.1 Functions and Graphs
1.4.1.1 Linear Function
Linear functions have the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. The following figure plots \(f(x) = 83x + 250\):
Consider line \(L\) passing through points \((x_1, y_1)\) and \((x_2, y_2)\). Let \(\triangle{y} = y_2 - y_1\) and \(\triangle{x} = x_2 - x_1\) denote the changes in \(y\) and \(x\), respectively. The slope of the line is:
1.4.1.2 Polynomials
A linear function is a special type of a more general class of functions: polynomials. A polynomial function is any function that can be written in the form:
for some integer \(n \ge 0\) and constants \(a_n, a_{n-1}, \cdots, a_0\), where \(a_n \ne 0\).
1.4.1.3 Trigonometric Functions
1.4.1.4 Exponential and Logarithic
1.4.2 Limits
1.4.3 Derivatives
1.4.3.1 Defining the Derivative
Now that we have both a conceptual understanding of a limit and the practical ability to compute limits, we have established the foundation for our study of calculus, the branch of mathematics in which we compute derivatives and integrals.
Tangent Lines
We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point \((a, f(a))\) to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of \(x\) near \(a\) and drawing a line through the points \((a, f(a))\) and \((x, f(x))\). The slope of this line is given by an equation in the form of a difference quotient:
We can also calculate the slope of a secant line to a function at a value a by using this equation and replacing \(x\) with \(a + h\), where \(h\) is a value close to 0. We can then calculate the slope of the line through the points \((a, f(a))\) and \((a + h, f(a + h))\). In this case, we find the secant line has a slope given by the following difference quotient with increment \(h\):
These two expressions for calculating the slope of a secant line are illustrated in Figure 1. We will see that each of these two methods for finding the slope of a secant line is of value. Depending on the setting, we can choose one or the other. The primary consideration in our choice usually depends on ease of calculation.
The Derivative of a Function at a Point
1.4.3.2 The Derivative as a Function
1.4.3.3 Differentiation Rules
1.4.3.4 Derivatives as Rates of Change
1.4.3.5 Derivatives of Trigonometric Functions
1.4.3.6 The Chain Rule
We have seen the techniques for differentiating basic functions (\(x^n\), \(sinx\), \(cosx\), etc.) as well as sums, differences, products, quotients, and constant multiples of these functions. However, these techniques do not allow us to differentiate compositions of functions, such as \(h(x) = sin(x^3)\) or \(k(x) = \sqrt{3x^2 + 1}\). In this section, we study the rule for finding the derivative of the composition of two or more functions.