Lesson 1.2 Calculus

Mathematical model:

A mathematical description of a real world situation.

Mathematical models are often represented as functions.

These functions may be linear, quadratic, cubic, etc.

A linear function of x means the graph of the function is a line.

A linear function of x means the graph of the function is a line. Also the equation could be put in the form of f(x) = mx + b where m = slope and b = y-intercept.

A characteristic of linear functions is that they grow at a constant rate.

For example, the linear function: f(x) = 3x – 2

Notice from the table, as x increases by 0.1, the value of f(x) increases by 0.3. We already know that the slope of the graph is 3, but that is also interpreted as the constant rate of change.

Example:

(a) As dry air moves upward, it expands and cools. If the ground temperature is 20 0 C and the temperature at a height of I km is 10 0 C, express the temperature T(in 0 C) as a function of the height h (in km), assuming that a linear model is appropriate.

Example: (b) Draw the graph of the function in part (a). What does the slope represent?

Example: (b) Draw the graph of the function in part (a). What does the slope represent?

Example: (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km?

Polynomials:

A function P is called a polynomial if where n is a nonnegative integer and the numbers a 0, a 1, a 2, etc are constants, called the coefficients.

Polynomials: The degree of a polynomial is the term with the largest exponent (or sum of exponents if more than one variable exists within the term).

Polynomials: The degree of a polynomial is the term with the largest exponent (or sum of exponents if more than one variable exists within the term).

Polynomials: The degree of a polynomial is the term with the largest exponent (or sum of exponents if more than one variable exists within the term). The above polynomial has degree 6.

Examples of quadratic functions: P(x) = ax 2 + bx + c

Examples of cubic functions: P(x) = ax 3 + bx 2 + cx + d

Power Functions:

A function in the form of f(x) = x a.

Power Functions: A function in the form of f(x) = x a. If a = n is a positive integer then,

Power Functions: A function in the form of f(x) = x a. If a = n is a positive integer then,

Power Functions: A function in the form of f(x) = x a. If a = n is a even then,

Power Functions: A function in the form of f(x) = x a. If a = n is a even then,

Power Functions: A function in the form of f(x) = x a. If a = n is a odd then,

Power Functions: A function in the form of f(x) = x a. If a = n is a odd then,

Power Functions: If a = 1/n, where n is a positive integer: f(x) = x 1/n = ?? (This is called the root function)

Power Functions: If a = 1/n, where n is a positive integer: f(x) = x 1/n = ?? (This is called the root function)

Power Functions: If a = -1, we get what is called the reciprocal function xy = 1  y = 1/x

Power Functions: If a = -1, we get what is called the reciprocal function xy = 1  y = 1/x

Rational Functions:

A function that is a ratio of 2 polynomials.

Rational Functions: A function that is a ratio of 2 polynomials.

Rational Functions: A function that is a ratio of 2 polynomials.

Algebraic Functions: Any function that can be constructed using algebraic operations (addition, subtraction, multiplication, division, and taking roots).

Algebraic Functions: Any function that can be constructed using algebraic operations (addition, subtraction, multiplication, division, and taking roots). Any rational function is automatically an algebraic function.

Algebraic Functions: Here are some examples:

Algebraic Functions: Here are some examples:

Trigonometric Functions:

In Calculus, it is always assumed that radian measure is always used.

Trigonometric Functions: In Calculus, it is always assumed that radian measure is always used. We should recognize the following graphs:

Trigonometric Functions: In Calculus, it is always assumed that radian measure is always used. We should recognize the following graphs:

Trigonometric Functions: In Calculus, it is always assumed that radian measure is always used. We should recognize the following graphs:

Trigonometric Functions: The tangent function is related to the sine and cosine functions as:

Trigonometric Functions: The tangent function is related to the sine and cosine functions as:

Trigonometric Functions: The tangent function is related to the sine and cosine functions as:

Trigonometric Functions: The tangent function is related to the sine and cosine functions as: Its Range is:

Trigonometric Functions: The tangent function is related to the sine and cosine functions as: Its Range is: Its Domain is:

Exponential Functions:

where the base a is a positive constant. f(x) = a x

Exponential Functions: where the base a is a positive constant. f(x) = a x

Exponential Functions: where the base a is a positive constant. Its Range is: f(x) = a x

Exponential Functions: where the base a is a positive constant. Its Range is:Its Domain is: f(x) = a x

Logarithmic Functions:

f(x) = log a x

Logarithmic Functions: f(x) = log a x

Logarithmic Functions: Its Range is: f(x) = log a x

Logarithmic Functions: Its Range is:Its Domain is: f(x) = log a x