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QuestionFind the slope from (3,f(3))(3, f(3)) to (3+h,f(3+h))(3+h, f(3+h)) for f(x)=x2+9xf(x)=x^{2}+9x as a function of hh.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x+9xf(x)=x^{}+9 x . We are finding the slope of the line from (3,f(3))(3, f(3)) to (3+h,f(3+h))(3+h, f(3+h))

STEP 2

The slope of a line between two points (x1,y1)(x1, y1) and (x2,y2)(x2, y2) is given by the formulalope=y2−y1x2−x1lope = \frac{y2 - y1}{x2 - x1}

STEP 3

In this case, our two points are (3,f(3))(3, f(3)) and (3+h,f(3+h))(3+h, f(3+h)). So we can write the slope aslope=f(3+h)−f(3)(3+h)−3lope = \frac{f(3+h) - f(3)}{(3+h) -3}

STEP 4

implify the denominator of the slope formulalope=f(3+h)−f(3)hlope = \frac{f(3+h) - f(3)}{h}

STEP 5

Now, we need to find the values of f(3+h)f(3+h) and f(3)f(3) by substituting 3+h3+h and 33 into the function f(x)f(x) respectively.

STEP 6

Find f(3+h)f(3+h)f(3+h)=(3+h)2+9(3+h)f(3+h) = (3+h)^{2}+9(3+h)

STEP 7

Expand the expressionf(3+h)=9+6h+h2+27+9hf(3+h) =9+6h+h^{2}+27+9h

STEP 8

implify the expressionf(3+h)=h2+15h+36f(3+h) = h^{2}+15h+36

STEP 9

Find f(3)f(3)f(3)=32+9(3)f(3) =3^{2}+9(3)

STEP 10

Calculate the valuef(3)=9+27=36f(3) =9+27 =36

STEP 11

Now, substitute f(3+h)f(3+h) and f(3)f(3) into the slope formulalope=h+15h+36−36hlope = \frac{h^{}+15h+36 -36}{h}

STEP 12

implify the expressionlope=h2+15hhlope = \frac{h^{2}+15h}{h}

STEP 13

Finally, we can simplify this expression by dividing each term in the numerator by hhlope=h+15lope = h+15So, the slope of the line from (3,f(3))(3, f(3)) to (3+h,f(3+h))(3+h, f(3+h)) is h+15h+15.

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