Math

QuestionFind f(a)f(a), f(a+h)f(a+h), and the difference quotient f(a+h)f(a)h\frac{f(a+h)-f(a)}{h} for f(x)=67xf(x)=6-7x where h0h \neq 0.

Studdy Solution

STEP 1

Assumptions1. The function is given as f(x)=67xf(x) =6 -7x . We are asked to find f(a)f(a), f(a+h)f(a+h), and the difference quotient f(a+h)f(a)h\frac{f(a+h)-f(a)}{h}
3. h0h \neq0 to avoid division by zero in the difference quotient

STEP 2

First, we need to find f(a)f(a). We can do this by substituting x=ax = a into the function.
f(a)=67af(a) =6 -7a

STEP 3

Now, we need to find f(a+h)f(a+h). We can do this by substituting x=a+hx = a + h into the function.
f(a+h)=67(a+h)f(a+h) =6 -7(a+h)

STEP 4

implify the expression f(a+h)f(a+h).
f(a+h)=67a7hf(a+h) =6 -7a -7h

STEP 5

Now, we need to find the difference quotient f(a+h)f(a)h\frac{f(a+h)-f(a)}{h}. We can do this by substituting the expressions we found for f(a)f(a) and f(a+h)f(a+h) into the difference quotient.
f(a+h)f(a)h=(7a7h)(7a)h\frac{f(a+h)-f(a)}{h} = \frac{( -7a -7h) - ( -7a)}{h}

STEP 6

implify the numerator in the difference quotient.
f(a+h)f(a)h=hh\frac{f(a+h)-f(a)}{h} = \frac{-h}{h}

STEP 7

implify the difference quotient by canceling out hh in the numerator and the denominator.
f(a+h)f(a)h=7\frac{f(a+h)-f(a)}{h} = -7So, f(a)=67af(a) =6 -7a, f(a+h)=67a7hf(a+h) =6 -7a -7h, and the difference quotient f(a+h)f(a)h=7\frac{f(a+h)-f(a)}{h} = -7.

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