I'm not sure how to do this problem. What does the Δ mean and how do you solve it?

Question

Find:

f(x+Δx) - f(x) / Δx if f(x)= 8x^2+1

Answer 1

Δx means "change in x."

What you have there is called a difference quotient. It is the slope of the line between two points. The points are on the curve f(x) and can be written:

( x , f(x) )
( x+Δx , f(x+Δx) )

So the change in x is given:

(x+Δx) - x = Δx

The change in y is:

f(x+Δx) - f(x)

So the slope is the change in y, divided by the change in x:

( f(x+Δx) - f(x) ) / Δx

For an image, and similar explanation, see:
http://en.wikipedia.org/wiki/Secant_line#Secant_approximation

So for yours, let's work it out. First simplify the numerator of the fraction:

f(x+Δx) - f(x)

= [ 8(x+Δx)² + 1 ] - [ 8x² +1 ]

= 8(x+Δx)² - 8x² +1 - 1

= 8x² + 16x(Δx) + 8(Δx)² - 8x²

= 8x² - 8x² + 16x(Δx) + 8(Δx)²

= 16x(Δx) + 8(Δx)²

Now divide through by Δx for your answer:

16x(Δx) / Δx + 8(Δx)² /Δx

= 16x + 8Δx

That is your final answer.

The difference quotient is how you find derivatives too. The limit as Δx → 0 becomes the derivative whenever a derivative exists. However, it did NOT ask for the derivative, so you cannot ignore the 8Δx term or try to take the derivative some other way, shortcutting the difference quotient. But, if you wanted to take the derivative too, it is:

16x + 8(0) = 16x

Answer 2

Hi,

First of all, the symbol Δ (delta) represents a down-scaling of whatever comes next to it. Δx is "a very small amount of x". The whole expression f(x+Δx) - f(x) / Δx is no other than the "regular" and well-known derivative.
Formally, the derivative is the local slope of the function f(x). Because f(x) isn't always a linear function (such as f(x) = a*x +b), one must try to see whether he can build ANOTHER function, which will describe the slope AT EACH AND EVERY X. This is done by asking the basic question, which is actually also what happens with linear functions:

1) take a small step in the x direction - this is Δx
2) measure the value of f - by measuring f(x+Δx)
3) calculate how much did the function (y-axis) changed, relative to how much the x changed - that's exactly:
[f(x+Δx) - f(x)] / Δx

if you know some trigonometry, this would also be intuitively connected to "the slope", since the last expression is EXACTLY tan(alpha), where alpha is the angle that the slope makes with the x-axis. Note that the above expression depends on x, f(x) etc, and so varies in its VALUE from one point to another. However, it may be written down as a SINGLE functional form.

Now that we understood why this expression is the derivative, lets see how do we actually SOLVE this type of questions. I'd assume that you studied the idea & use of "limits" in calculus. lets begin then....

f(x+Δx) = 8(x+Δx)^2+1 = 8(x^2+2x Δx + (Δx)^2) + 1
this is a simple square expansion. now lets carry on:

f(x+Δx)-f(x) = [8(x^2+2x Δx + (Δx)^2) + 1] - [8x^2 +1]
= 8x^2 +16x Δx + 8(Δx)^2 +1 - 8x^2 -1
= 16x Δx + 8(Δx)^2

and so:
[f(x+Δx) - f(x)] / Δx = [16x Δx + 8(Δx)^2]/Δx
= 16x + 8Δx

and that's ALMOST the end. now, usually, the REAL question is to actually get to the derivative, by taking the limit Δx --> 0 of the original expression. In our case:

lim{Δx --> 0} ([f(x+Δx) - f(x)] / Δx) = lim{Δx --> 0} (16x + 8Δx)

= 16x
AND THIS IS THE FINAL ANSWER!
now, if you have any experience with "simple" derivatives, you'll be able to recognize this result, since the derivative of x^2 is 2x, and here everything was multiplied by 8....

Good Luck, Benny

Answer 3

delta x (triangle x) is a little bit of change in x.
Just substitute and do the algebra.
f(x + delta_x) = 8(x + delta_x)^2 + 1
= 8(x^2 + 2xdelta_x + delta_x^2) + 1
= 8x^2 + 16xdelta_x + 8delta_x^2. + 1
So, f(x + delta_x) - f(x) /delta_x = (8x^2 + 16xdelta_x + 8delta_x^2 + 1 - (8x^2 + 1))/delta_x
= (8x^2 + 16xdelta_x + 8delta_x^2 + 1 - 8x^2 - 1)/delta_x^2
= (16xdelta_x + 8delta_x^2)/delta_x
= 16x + 8delta_x
In this type of problem, all the terms in the numerator without delta_x should cancel out. If not, you have probably made a sign error.

Answer 4

Δ means difference, usually a small one that will shrink to 0 as a limit. replace it with h since that's easier to type. then

[f(x+h) - f(x)]/h = [8(x+h)² + 1 - (8x² + 1)]/h =
[8x² + 16xh + 8h² + 1 - 8x² - 1]/h =
[16xh + 8h²]/h =
16x + 8h

this "difference quotient" is the slope of the secant line between (x, f(x)) and (x+h, f(x+h)). when "in the limit" h shrinks to 0, what you have is the slope of the tangent to the graph of the function at (x, f(x)), in this case 16x. this is also the derivative of the function, if you're in calculus.

Answer 5

It is a delta (Greek letter). My calculus book just uses "h". The symbol means "change in x". What the expression involves is some distance (delta x) from a point (x). Really this equation is finding slope between an equation and a point on that equation. Use (h) where there is a delta and do algebra. It will simplify easily. Then you apply the limit and you should get 16x for the slope.

Answer 6

delta means the change in x
to solve this problem substitute x+deltax in the place of x in equation(8x^2+1) then aplly limit to find the answer
u also have a beter option
Differentiate th above equation(actually it's wat they ask)
after differentiating u get 16x and it's the answer
LHS means to differentiate