ECE Review: Functions and Domain

Functions and Relations

• A function is a mathematical relation that relates an input from a set called the domain to exactly one output in a set called the range.
• A function can be represented as a relation, equation, or graph.

Domain of Functions

• The domain of a function is the set of input values for which the function is defined.
• The domain of a function can be found by analyzing the function's formula.

Limits of Functions

• The limit of a function as x approaches a certain value is the value that the function approaches as x gets arbitrarily close to that value.
• The limit of a function can be determined using various techniques, such as factoring, canceling, or using trigonometric identities.

Piecewise Functions

• A piecewise function is a function that is defined differently for different intervals of the input variable.
• The limit of a piecewise function can be evaluated by considering the limit of each piece of the function separately.

Continuity of Functions

• A function is continuous at a number a if the function is defined at a and the limit of the function as x approaches a exists and is equal to the value of the function at a.
• The conditions for continuity of a function at a number a are:
  • f(a) is defined
  • limx→af(x)lim_{x \to a} f(x)limx→a​f(x) exists
  • limx→af(x)=f(a)lim_{x \to a} f(x) = f(a)limx→a​f(x)=f(a)

Calculus Problems

Problem 6: Limit of (2-x)tan(2) as x approaches 1

• The problem involves finding the limit of an algebraic function multiplied by a trigonometric function
• The function is (2-x)tan(2) and the limit is as x approaches 1
• Possible answers: A. e^2, B. e^2/π, C. 0, D. ∞

Problem 7: Limit of sin(2(x-π/4))/(x-π/4) as x approaches π/4

• The problem involves finding the limit of a trigonometric function
• The function is sin(2(x-π/4))/(x-π/4) and the limit is as x approaches π/4
• Possible answers: A. 0, B. 1, C. 2, D. ½

Problem 8: Evaluating a Limit

• The problem involves evaluating a limit of an integral
• The integral is ∫t+√(t^2+5)t 1/(x^2) dx and the limit is as x approaches 0
• Possible answers: A. 2, B. 3, C. 4, D. 5

Problem 9: Solving for A in a Piecewise Function

• The problem involves solving for a parameter in a piecewise function
• The function is f(x)={x-4, x and the goal is to solve for A
• No possible answers are provided