Calculus problem involving secant line and tangent line.

[thatguy47]

I'm stuck right now. Here's the problem:

I'm given the function f(x) = "under radical"(1 + x) and I had to sketch a graph of that.

This is the graph I get for that:


Next I have to Sketch the secant line to f between the points with x-coordinates x=2 and x=4 and I did that below (hopefully correct):



The third step is the one I'm stuck on. It's: Sketch the secant lines to f between the pairs of points with the following x-coordinates, and compute their slopes:
a) x=2 and x=3
b) x=3 and x=4
c) x=2.5 and x=3.5
d) x=2.8 and x=3.2

This is what I get when I "Sketch the secant lines to f between the pairs of points with the following x-coordinates"



Then I compute their slopes...
a)(2, rad3) (3,2)
(2-rad3)/(3-2) equals .268

b)(3,2) (4, rad5)
(rad5 - 2)/(4-3) equals .236

c) (2.5, 1.87) (3.5, 2.121)
(2.121 - 1.87)/(3.5 - 2.5) equals .251

d) (2.8, 1.949) (3.2, 2.049)
(2.049 - 1.949)/(3.2 - 2.8) equals .25

Next step....Using the slopes you've found so far, estimate the slope of the tangent line at x = 3

This is where I'm stuck at. How do the estimate the slope of the tangent line?

The next step is the same, except estimate for x = 8

The final question if, based on the last two estimates, guess the slope of the tangent line at any point x = a, for a > -1.

I'm stuck. Please help me if you know how to finish this. Thanks.

 

[tkhunny]

What an odd question. That part you are stuck on? You just did it five times.

1) The slope between 2 and 4 might be a good estimate of the slope at 3. You see yourself how close the secant is to the function.
2) The slope between 2 and 3 might be a good estimate of the slope at 3, but ignores information on the right.
3) The slope between 3 and 4 might be a good estimate of the slope at 3, but ignores information on the left.
4) The slope between 2.5 and 3.5 might be a better estimate than [2,4], since it ignores information farther from 3.
5) The slope between 2.8 and 3.3 might be a better estimate than [2.5,3.5], since it ignores information farther from 3.

So, how many estimates do you want? Can you see that we may have defined a limiting process for determining the slope AT 3?

 

[thatguy47]

Thanks for the help...

When it told me to sketch the secant line to f between the points with x-coordinates between x=2 and x=4 did I do that right?

Same with the other 4:

I think I sketched it wrong.

For the part I said I was stuck on:
4. Using the slopes you've found so far, estimate the slope of the tangent line at x = 3.
Could the answer just be .25? The one I got from part d.

5. Repeat problem 4 for x=8.
I'm still confused.... The tangent line slope would be 3/8?

And for the last questions:
6. Based on Problems 4 and 5, guess the slope of the tangent line at any point x = a, for a > -1.
would this just be .3? I'm still really lost. Sorry.

 

[tkhunny]

This curve is so flat, nearly any reasonable approximation will be very close.

From my list of 5

1) Central approximation with range of 2. Good approximation in this case.
4) Central approximation with range of 1. Very Good approximation in this case.
5) Central approximation with range of 0.4. Really Very Good approximation in this case.

The other two might need an average to produce a better approximation.
2) The slope between 2 and 3 might be a good estimate of the slope at 3, but ignores information on the right.
3) The slope between 3 and 4 might be a good estimate of the slope at 3, but ignores information on the left.

For approximating x = 8, do the same thing.

Slope of secant on [7,9]
Slope of secant on [7.5,8.5]
Slope of secant on [7.9,8.1]

You should get better and better results. It's even flatter at x = 8 than it was at x = 3, so they all will be pretty close.

 

[thatguy47]

Thanks, I see what you are saying now:

Here are my answers:

4. Using the slopes you've found so far, estimate the slope of the tangent line at x = 3.
I just used the estimate from part d (.25)

5. Repeat problem 4 for x=8.
I found the slope between 7.9 and 8.1 which was .17

And for the last question:
6. Based on Problems 4 and 5, guess the slope of the tangent line at any point x = a, for a > -1.
I guessed .3. Hopefully thats close.

One last question: Did my graph look right for sketching the secant lines? Thanks for all of your help !

 

[tkhunny]

Are you sure it want's a single, numeric answer? The slope for x < 0 isn't nearly as flat as the slope near x = 8.

i was thinking something like this: \(\displaystyle \frac{f(a+0.1)-f(a-0.1)}{0.2}\)