3.11 Linear Approximations and Differentials
It is possible to calculate the value of f(a) of a function where the graph of f(x) is tangent at the point a, f(a) but calculating for any values that are extremely near a is difficult.
the equation of the linear approximation/tangent line approximation of f at a is
L(x) = f(a) + f '(a)(x - a)
How to get that equation:
Remember the formula f '(a) = (f(x) - f(a))/(x - a) ?
solve for f(x). . .and voila!
you get f(x) = f(a) + f '(a)(x - a). ^_^
For example, suppose the murderer from Ismael's project group's movie wants to bake his victim for dinner (okok, we don't know exactly if he really is a cannibal, but let's just assume that he is one). The temperature of the body is 36 degrees C. Suppose it is baked at 187 degrees C. After an hour, the body is 52 degrees C and after 2 hours it is 66 degrees C. (not 666, haha)
Predict the temperature of the body after four hours. (If you get this right, you won't be the next victim - just kidding)
The equation could be set up with f (x) as the temperature of the body after x hours. Therefore,
f(0) = 36
f(1) = 52
f(3) = 66
we need to find f '(2) in order to make a linear approximation.
f '(2) = lim x->2 (f(x) - f(2))/(x-2)
we can use x = 1 here. (at one hour) so,
f '(2) = (f(1) - f(2))/(1-2) = (52-66)/(-1) = 14 degrees C/hour
this answer is the instantaneous rate of temprature change by the average rate of change between 1 and 2 hours. therefore, the linear approximation is f(4) = f(2) + f'(2)(4-2) = 52+14(2) = 80.
another (more accurate) method is to plot the givein data and estimate the slope of the tangent line (p. 263 of your textbook)
*Tip: when a question asks you to find the linearization of a function which has radicals in it, always change the radical into an exponent (p. 263)
Differentials: If y = f(x) and f(x) is differentiable, then the differential dx is an independent variable. (dx can be any real number) the differential dx is defined by:
dy = f '(x) dx
Refer to the illustration on p. 265.
dx is the change in x from the point at which the tangent line is tangent to another point. dx is the change x.
dy, however is NOT just the change in y.
suppose you are given the points (x1, y1) and (x2, y2). on the original graph. the graph is tangent at (x1, y1) but at x2 the value of the tangent line is b. dx is the change in x from x1 to x2. BUT! change in y is the change from y1 to y2. dy is the change from y1 to b.
Well, hope that helped...and here ends the rant of the evil math student ~ OY!!! JEFF YOU'RE UP NEXT! *poke, poke* =)
What I think about while I play online games (studying for calculus :D)