finding the slope of the secant line for x^2 - 3x - 4

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Math wiz ya rite 09

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Hello! I bought an SAT practice book to get ready for the big test. It is pretty well formated, except it only has the answers/explanations to random questions. I was just hoping that you guys could help me with a question here or there that I didn't know how to solve. Thanks!

Let f(x) = x^2 - 3x - 4.

a.) Find the slope, m(h), of the secant line between the points A(1, f(1)) and B(1 + h, f(1 + h)).

*** I think it should be in the terms of h, but I'm not sure...?

b.) Evaluate m(0.01).

*** I believe this is an approximation of the slope of the line tangent to the curve at (1, f(1)).
 
a) Yes, the slope formula would be in terms of "h".

b) You are correct. But this exercise is in the context of algebra, so follow the instructions; don't take the derivative.

Eliz.
 
Re: finding the slope

Math wiz ya rite 09 said:
Hello! I bought an SAT practice book to get ready for the big test. It is pretty well formated, except it only has the answers/explanations to random questions. I was just hoping that you guys could help me with a question here or there that I didn't know how to solve. Thanks!


Let f(x) = x^2 - 3x - 4.

a.) Find the slope, m(h), of the secant line between the points A(1,f(1)) and B(1+h,f(1+h)).
***I think it should be in the terms of h, but not sure?

b.) Evaluate m(0.01).
***I believe this is an approximation of the slope of the line tangent to the curve at (1,f(1)).
Click to expand...

f(1) = 1<SUP>2</SUP> - 3(1) - 4
f(1) = -6
So, A is the point (1, -6)

f(1 + h) = (1 + h)<SUP>2</SUP> - 3(1 + h) - 4
f(1 + h) = 1 + 2h + h<SUP>2</SUP> - 3 - 3h - 4
f(1 + h) = h<SUP>2</SUP> - h - 6
So, B is the point (1 + h, h<SUP>2</SUP> - h - 6)

Use these coordinates to find the slope....

(y<SUB>2</SUB> - y<SUB>1</SUB>) / (x<SUB>2</SUB> - x<SUB>1</SUB>)

(h<SUP>2</SUP> - h - 6 - (-6)) / (1 + h - 1)

(h<SUP>2</SUP> - h) / h

Factor the numerator and reduce the fraction....

This is m(h)....

Now, to find m(.01), substitute .01 for h. And you're right; this would be an approximation for the slope of the tangent to the curve at x = 1.
 
Re: finding the slope

Math wiz ya rite 09 said:
Ok so i factor (h<SUP>2</SUP> - h) / h to get [(h-1)(h+1)] / (h)

Now what?
Click to expand...

uhh ...

(h<sup>2</sup> - h)/h = h(h - 1)/h = h - 1
 
no, the slope is a function of h ... m(h) = h - 1

m(0.01) = 0.01 - 1
 
Re: finding the slope

I understand m(0.01) and how it equals 0.01 - 1 which equals -0.99, but I still am not understanding the slope part of the question.What is y1 y2 x1 & x2?
 
Re: finding the slope

Math wiz ya rite 09 said:
I understand m(0.01) and how it equals 0.01 - 1 which equals -0.99, but I still am not understanding the slope part of the question.What is y1 y2 x1 & x2?
Click to expand...

you were given ...

f(x) = x<sup>2</sup> - 3x - 4

you were asked to find the slope of the secant line between A (1, f(1))
and B (1+h, f(1+h))

(y<sub>2</sub> - y<sub>1</sub>)/(x<sub>2</sub> - x<sub>1</sub>) = [f(1+h) - f(1)]/[(1+h) - 1]

using h = 0.01 ...

y<sub>2</sub> = f(1.01)
y<sub>1</sub> = f(1)
x<sub>2</sub> = 1.01
x<sub>1</sub> = 1

but that's not the point ... you completed this exercise to find an equation for the slope of the secant line for any value of h, not just 0.01.

As h gets smaller and smaller (approaches 0), the slope of the secant line approaches a limit ... the slope of a tangent line to the curve at x = 1, also called the instantaneous rate of change of f(x) at x = 1. In other words, you just did a preliminary exercise involving differential calculus.
 
Math wiz ya rite 09 said:
im only a freshman and although i may be in precalc, this problem is has my mind fryed!
Click to expand...
In that case you need a seat down with your academic advisor.
See if you are really in the correct course.
You know that that you may not correctly placed.
 
no reason to get p.o.'ed there, Zeke ...

the slope is -0.99.

you're welcome.
 
Ok......I did all of the "hard stuff" for you.

And someone else showed you how to simplify
(h^2 - h) / h

to

h - 1

that is m(h).....
m(h) =h - 1

So, to find m(0.01), simply subistitute 0.01 for h:

m(0.01) = 0.01 - 1

If you got lost in that, please tell us WHERE you got lost....
 
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