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Evaluating Functions and Difference Quotient

Evaluating Functions and Difference Quotient. Objectives. Standard 24.0 I will evaluate the value of any functions, including piecewise functions. Calculus Standard 4.0 I will evaluate the difference quotient of any given function.

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Evaluating Functions and Difference Quotient

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  1. Evaluating Functions and Difference Quotient

  2. Objectives Standard 24.0 • I will evaluate the value of any functions, including piecewise functions. Calculus Standard 4.0 • I will evaluate the difference quotient of any given function.

  3. When we evaluate a function we are finding the function value for a specific input. To do this we replace the function variable in the function’s formula with the specific input and proceed from there. The “specific input” can be a constant, another variable or an algebraic expression. The important thing to remember is it replaces the function variable everywhere in the function’s formula For example:

  4. Another example: Because is an imaginary number 1 is not in the domain of f (x)

  5. Another example:

  6. Another example: Because 5/0 is undefined 4 is not in the domain of h(x).

  7. Problems - 1 Given find

  8. Problems - 2

  9. Sometimes a function has different rules or formulas depending on what the input value is. These functions are known as piece-wise definedfunctions.

  10. Problems - 3

  11. The Difference Quotient The difference quotientof a function f (x) is defined as follows: This is used in calculus when finding derivatives so it is worthwhile to become familiar with it in precalculus.

  12. As we take Q closer to P, the accuracy with which the slope of the secant line approximates the slope of the tangent line increases.

  13. Find the difference quotients for the following functions:

  14. The difference quotient for

  15. The difference quotient for

  16. The difference quotient for

  17. Problems - 4 Find the difference quotient for the following function (click on mouse to see answer).

  18. Problems - 5 Find the difference quotient for the following function (click on mouse to see answer).

  19. Practice Textbook P.69 Q.73-80 Answers

  20. Problems - 6 Find the difference quotient for the following function (click on mouse to see answer).

  21. Average Rate of Change A special kind of difference quotient is the average rate of change.We can use the function values at two different points, a and b to find the average rate of change of a function over the interval [ a, b ]. This is given by: Notice, this is equal to the slope of the line connecting the two points ( a, f(a) ) and ( b, f(b) ). The average rate of change for the function f (x) = x2 over the interval [2,4] is:

  22. Find the average rate of change for each of the following functions over the given intervals (click on mouse to see answer).

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