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Calculus/Change in Y on the tangent line

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Expert: - 8/1/2011


Question
Given: The curve is y=f(x), x starts out with a and then changes by the infinitesimal amount dx.

What happens to y along the curve and along the tangent line?

I understand along the curve that the change in y is:

Dy = f(a + dx) – f(a)

But I am completely lost when it comes to the tangent line. Any help would be appreciated, thanks,

Aaron

Answer
The tangent line gives the slope of the curve at that point.
For example, if y = x², then it can be said that dy/dx = 2x.
So at the point x=1, we have y=1²=1 and the slope is 2*1 = 2;
at the point x=2, we have y=2²=4 and the slope is 2*2=4;
at the point x=3, we have y=3²=9 and the slope is 2*3=6.

The proper formula is dy/dx = lim(a->0){[f(x+a)-f(x)]/a}.
If y = x^n, dy/dx = ny^(n-1).  This dy/dx is also called y'.
Thus, if y = 3x² + 7x + 11, that can be rewritten as y = 3x^2 + 7x^1 + 11.
From this, it can be seen that dy/dx = 2*3x^(2-1) + 7x^(1-1) = 6x^1 + 7x^0 = 6x + 7.
Thus, at x = 2, y = 12+14+11 = 37 and
the slope of the tangent line at that point is 6*2+11 = 23.

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