Calculus

• A rational expression is a fraction with variables in it, like x + 2 / x^2 - 3
• To simplify a fraction to lowest terms, factor the numerator and denominator, then cancel common factors
• Multiplying two fractions with numbers: multiply numerators and denominators
• Dividing two fractions: multiply by the reciprocal of the fraction in the denominator
• To add or subtract rational expressions, find a common denominator and perform the operation on the numerators
• The process for finding the sum of two rational expressions with variables is similar to adding fractions with numbers
• The average rate of change for a function on an interval is the slope of the secant line between two points on the graph
• The difference quotient represents the average rate of change of a function on an interval using x and x + h notation
• The difference quotient is the slope of the secant line for the graph of a function between two points
• To rewrite the average rate of change as f(x + h) - f(x) / h, we first substitute x + h into the function formula
• Simplifying the expression, we get 4xh + 2h^2 - h
• The difference quotient formula is (4x + 2h - 1)
• In calculus, the difference quotient is used to find the derivative or slope of a function
• The concept of limits is introduced through graphs and examples
• Limits are used to describe the behavior of functions as they approach a certain value
• One-sided limits exist when the function approaches a value from either the left or right side
• Infinite limits can occur when the function approaches positive or negative infinity
• Limits may not exist if the function approaches different values from the left and right sides
• The behavior of functions near specific values can be analyzed using limits
• Limits can fail to exist if the function approaches different values from different directions or exhibits wild behavior
• Limits can fail to exist due to wild behavior, where the function does not settle down at any single value
• In the case of wild behavior, the limit does not exist because the function's values oscillate between different numbers infinitely often as x approaches a certain value
• Limit laws include rules for finding limits of sums, differences, products, and quotients of functions
• One of the limit laws states that the limit of the sum of two functions is equal to the sum of their individual limits
• Similarly, the limit of the difference of two functions is equal to the difference of their individual limits
• The limit of a constant times a function is equal to the constant multiplied by the limit of the function
• The limit of the product of two functions is equal to the product of their individual limits
• The limit of the quotient of two functions is equal to the quotient of their individual limits, provided that the limit of the denominator is not zero
• When using limit laws, it is important that the limits of the component functions exist as finite numbers
• If the limits of the component functions do not exist, then the limit laws cannot be applied, and other techniques must be used to evaluate the limit
• The Squeeze Theorem, also known as the pinching theorem or the sandwich theorem, is a method for finding limits when a function is trapped between two other functions with the same limit
• In the Squeeze Theorem, if a function is squeezed between two other functions with the same limit as x approaches a certain value, then the squeezed function also has the same limit at that value
• The Squeeze Theorem is useful for evaluating limits when a function is bounded by two other functions with known limits
• When evaluating limits, if both the numerator and denominator approach zero, simplifying the expression by factoring or multiplying out can help
• In the first example, the limit is calculated by factoring the expression and simplifying it before plugging in the value
• The second example involves multiplying out the expression to simplify it before evaluating the limit
• In the third example, adding together rational expressions helps simplify the expression before finding the limit
• The fourth example uses the conjugate trick to simplify the expression before evaluating the limit
• The fifth example involves using cases and looking at one-sided limits to evaluate the expression when the limit does not exist
• Different methods like factoring, multiplying out, adding rational expressions, using the conjugate trick, and considering cases can be used to evaluate limits