Suppose that f (x) is a function such that the relationship given below is true.

f (3 + h) - f (3) = 9h^2 + 8h

What is the slope of the secant line through (3, f (3)) and (7, f (7))?

I am stuck on this one , tried every thing in the world. i don't know how to do it since there is no f(x) given.. so please be descriptive if you help. This is due tomorrow that's why i am posting it again, don't mean to spam.

please show your work. maybe myself or the other tutors could figure out what you did wrong.

ok. this question also had a part a) which was


(a) What is f '(3)?
so i just divided
(9h^2+8h)/h

(h(9h+8))/h and plugged in 0 since lim->0

got 8 as my answer.

now for B)

i tried plugging 3 into (h) in func 9h^2+8h/h

did same with 7 and got wrong answer. i tried many different ways but kept getting wrong answer.

what i am confused is with how to find values for f(7) and f(3) since there is no function given to begin with...

There is no limitation on the value/size of h, so h can be any number.

f (3 + h) - f (3) = 9h^2 + 8h
so
f(3+h) = 9h^2 + 8h - f (3)
Try setting h=4 to see what you get.

Sorry, the equation should read:

f(3+h) = 9h^2 + 8h + f (3)

In fact, the solution is simpler than that.

Remember TutorCat said:
for the secant:
[f(7)-f(3)]/(7-3)
you can work out
[f(7)-f(3)]
from
f (3 + h) - f (3) = 9h^2 + 8h
by putting h=?

To find the slope of the secant line between two points on a curve, you need the values of the function at those points. Although the exact form of the function f(x) is not given, we can use the relationship provided to calculate the values of f(3) and f(7) using substitution.

From the given relationship, we have:

f(3 + h) - f(3) = 9h^2 + 8h

Let's substitute x = 3 + h:

f(x) - f(3) = 9h^2 + 8h

Now, let's substitute x = 7:

f(7) - f(3) = 9h^2 + 8h

At this point, we have two equations:

1) f(x) - f(3) = 9h^2 + 8h
2) f(7) - f(3) = 9h^2 + 8h

We want to find the slope of the secant line between (3, f(3)) and (7, f(7)). The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

slope = (y2 - y1) / (x2 - x1)

In this case, the points are (3, f(3)) and (7, f(7)). We have determined that f(3) and f(7) are both equal to (9h^2 + 8h). Substituting these values into the slope formula:

slope = (f(7) - f(3)) / (7 - 3)

slope = ((9h^2 + 8h) - (9h^2 + 8h)) / 4

The numerator cancels out, leaving us with:

slope = 0 / 4

Hence, the slope of the secant line between the points (3, f(3)) and (7, f(7)) is 0.