Final answer:
The difference quotient of the function f(x) = 13/4 - (5/4)x for nonzero values of h is calculated by substituting x + h into f(x), subtracting f(x), and dividing by h. The simplified expression of the difference quotient for this function is -5/4.
Step-by-step explanation:
The difference quotient of a function is used to approximate the slope of the tangent line to the function at a particular point and is a fundamental concept in calculus. For a function f(x), the difference quotient is given by the expression (f(x + h) - f(x))/h, where h is a nonzero value that approaches zero. In this case, for f(x) = ⅓ - (5/4)x, you would first calculate f(x+h) which becomes ⅓ - (5/4)(x + h). Then you subtract f(x) from this expression and divide by h to find the difference quotient.
To represent the difference quotient of f(x) = ⅓ - (5/4)x we follow these steps:
Substitute x + h into f(x) to compute f(x + h), which results in ⅓ - (5/4)(x + h).
Subtract f(x) from f(x + h) to get the numerator of the difference quotient: (⅓ - (5/4)(x + h)) - (⅓ - (5/4)x).
Divide the resulting expression by h to complete the difference quotient, which simplifies the expression to -5/4.