Lesson Derive (Calculus)

Source code of 'Derive (Calculus)'

This Lesson (Derive (Calculus)) was created by by Nate(3500)

In calculus, mathematicians try to derive the slope of a tangent (a line that hits a function at a certain arbitrary point). This includes any type of function (be it sine, cosine, tangent, parabolic, quintic ... technically, there are infinite amount of tangents to a linear line).
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~ Tangent Slope ~
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*Sketching a curve and drawing up points is heavily promoted for understanding reasons...
P(x,f(x)) and P(x + h,f(x + h)) where {{{h}}} is a value
Slope = (y2 - y1)/(x2 - x1)
{{{Slope = (f(x + h) - f(x))/(x + h - x)}}}
{{{Slope = (f(x + h) - f(x))/(h)}}} where {{{h}}} doesn't equal zero
*Let's apply this ....
Function: f(x) = 0.5x^2
Slope at P(2,2)? ...
{{{graph(600,400,-.5,3,-.5,5,0.5x^2,3x-4,2.5(x-2)+2,(10.5/3)(x-2)+2,2(x-2)+2)}}}
{{{h}}} was the greatest for the purple line
{{{h}}} decreased for the green line
{{{h}}} decreased further for the blue line
You can see: as {{{h}}} approaches zero ... a slope for a tangent is found
{{{h}}} approaches zero for the golden line
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~ Tangent Slope ~
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Slope of Tangent is written as = lim[h->0] {{{(f(x + h) - f(x))/(h)}}}
This can be broken down further by function: f(x) = x^r
lim[h->0] {{{(f(x + h) - f(x))/(h)}}}
lim[h->0] {{{((x + h)^r - x^r)/(h)}}}
lim[h->0] [C(r,0)(x^r)(h^0) + C(r,1)(x^(r-1))(h^1) ... + C(r,r)(x^0)(h^r) - x^r]/h
lim[h->0] [C(r,1)(x^(r-1))(h^1) ... + C(r,r)(x^0)(h^r)]/h
Which Simplifies:
lim[h->0] C(r,1)(x^(r-1)) ... + C(r,r)(h^(r-1))
Substitute:
lim[h->0] C(r,1)x^(r-1)
Simplify:
r*x^(r-1)
So, the slope of a tangent line to a function is expressed as r*x^(r - 1) (or f'(x) = r*x^(r - 1)) from the original function f(x) = x^r.
*Example:
f(x) = 0.5x^2 at P(2,2)
f'(x) = (0.5)(2)x^(2 - 1) = x
f'(2) = x = 2
The slope would be 2.
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This rule can not be applied to trig functions ... so, more work!
f(x) = sin(x)
lim[h->0] {{{(f(x + h) - f(x))/(h)}}}
lim[h->0] {{{(sin(x + h) - sin(x))/(h)}}}
lim[h->0] {{{(sin(x)cos(x) + sin(h)cos(x) - sin(x))/(h)}}}
lim[h->0] {{{(sin(x)cos(x) - sin(x))/(h)}}} + lim[h->0] {{{sin(h)cos(x)/h}}}
sin(x)*lim[h->0] {{{(cos(x) - 1)/(h)}}} + cos(x)*lim[h->0] {{{sin(h)/h}}}
sin(x)*0 - cos(x)*1
f'(x) = cos(x)
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f(x) = cos(x)
lim[h->0] {{{(f(x + h) - f(x))/(h)}}}
lim[h->0] {{{(cos(x + h) - cos(x))/(h)}}}
lim[h->0] {{{(cos(x)cos(h) - sin(x)sin(h) - cos(x))/(h)}}}
lim[h->0] {{{(cos(x)cos(h) - cos(x))/(h)}}} - lim[h->0] {{{sin(x)sin(h)/h}}}
cos(x)*lim[h->0] {{{(cos(h) - 1)/(h)}}} - sin(x)*lim[h->0] {{{sin(h)/h}}}
cos(x)*0 - sin(x)*1
f'(x) = - sin(x)
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Tangent slope and line at P(3.14,0) for f(x) = sin(x) ?
f'(x) = cos(x)
f'(3.14) = cos(3.14) = -1
y = -1(x - 3.14) + 0
y = -x + 3.14
{{{graph(600,300,-.5,5,-.5,1,sin(x),3.141592653-x)}}}