Question

I'm working through Session 1. I got the Sectant app to work and got all of the answers through it that I should have. My problem is that I can't get it to work out on paper. I'm going with the understanding that ΔY/ΔX should go like this
((.5x^3-x+ΔX)-x)/ΔX.

Answer 1

Not quite.
\[f \prime \left( x \right) = \lim_{\Delta x \rightarrow 0} \left[ \left( 0.5x^3 - x + \Delta x\right) - \left( 0.5x^3 - x) \right)\right]/\Delta x\]

Hope this helps.

Answer 2

The problem tells me that x = -0.75 and Δx= -0.5. When I run it [(0.5(-0.75)^3 −(-0.75)+-0.5)−(0.5(-0.75)^3 −(-0.75))]/-0.5

I get -2

Answer 3

Okay, let's look at this a different way, so as not to get bogged down in complex calculations. The red point is (-0.75, 0.539) and the yellow point is (-1.25, 0.273). Taking \[\Delta y / \Delta x\] and rounding to the nearest hundredth gives the correct answer.

Answer 4

Oh, and my original formula was wrong, sorry. The formula for the derivative is:
\[\lim_{\Delta x \rightarrow 0} [f(x + \Delta x) - f(x)]/ \Delta x\]This works out as \[\lim_{\Delta x \rightarrow 0} [((x + \Delta x)^3 - (x + \Delta x)) - (x ^3 - x)] / \Delta x \]

Answer 5

But remember, that formula is the derivative, which is not the same as \[\Delta y / \Delta x\]

Answer 6

To summarize my somewhat incoherent solution, delta y over delta x is the change in y over the change in x. I recommend using a calculator instead of just hovering your mouse, you need more accuracy. You do not need the formula for derivatives because the derivative is not the same as delta y over delta x, although delta y over delta x may approximate the derivative.

Also, to clarify, your derivative formula is a bit off. The formula for the derivative is \[\lim_{\Delta x \rightarrow 0} [f(x + \Delta x) - f(x)] / \Delta x\]which in this problem works out to be \[\lim_{\Delta x \rightarrow 0} [((x + \Delta x)^3 - (x + \Delta x)) - (x ^3 - x)]/ \Delta x\] and simplified makes \[\lim_{\Delta x \rightarrow 0} \Delta x^2 + 3x^2 + 3x \Delta x + 1 = 3x^2\]

I hope this helps. Let me know if you have any questions about how I worked everything out.

Answer 7

I've tried the formula you have given, using -.75 for x and -.5 for Δx, and still do not get the answer.

Answer 8

Are you still using the formula for the derivative? Remember, the derivative is not the same as delta y over delta x. I just fixed your formula for future reference. What you need is delta y over delta x. Do you know how to find that?

Answer 9

I guess I don't get any of what I'm trying to do. I can't get delta y over delta x and I'm not sure what I should be getting for an answer to the derivative.

Answer 10

Okay, let's start from the beginning. delta y over delta x is equal to the change in y over the change in x. y = f(x), so y at the red point is equal to x^3 - x, which is (-0.75)^3 - (-0.75). y at the yellow point is equal to x^3 - x, which is (-1.25)^3 - (-1.25). Subtract the y at the red point from the y at the yellow point. Subtract the x at the red point from the x at the yellow point. Then take the y-difference over the x-difference. If we let the red point be (x1, y1) and the yellow point be (x2, y2), we have \[(y2 - y1) / (x2 - x1)\] I hope this helps, and I'm proud of you for trying to work it out on paper instead of just relying on the mathlet.

Answer 11

Unless I'm completely crazy the answer should be 7.0625.

Answer 12

This answer doesn't seem to match up with anything on the app or the solution sheet.

Answer 13

Ok after taking a break and rechecking I finally got 0.53125. I need to get some sleep.