1
151 WebCalc Fall 2002-copyright Joe Kahlig
In Class Questions
MATH 151-Fall 02
October 17
What WebCalc calls The Differential Method is actually referred to by Stewart as
linear approximation or tangent line approximation of f at a.
f (x) ≈ f (a) + f 0 (a)(x − a)
If the formula is rewritten as L(x) = f (a) + f 0 (a)(x − a) then this called linearization of f at a.
Now Stewart’s definition of differentials is:
Let y = f (x), where f is a differentiable function. Then the differential dx is an independent variable: that is, dx can be given the value of any real number. The differential
dy is then defined in terms of dx by the equation dy = f 0 (x)dx
dy
This formula is found by starting with dx
= f 0 (x) and multiplying by dx.
The picture shows the differentials and relates them to ∆y and ∆x.
Now f (a + ∆x) = f (a) + ∆y. This can be
seen from the picture. Also note that as
∆x → 0 then dy gets closer to ∆y. Hence
f (a) + dy
f (a + ∆x)
∆y
dy
f (a)
f (a + ∆x) = f (a) + ∆y ≈ f (a) + dy
dx = ∆x
This new approximation is not different from
the tangent line approximation. (just takes a
little bit of algebra and realizing that x − a is
just dx).
a
a + ∆x
We can extend the approximations from lines to other polynomials.
The quadratic approximation to f (x) neat a is
f 00 (a)
f (x) ≈ f (a) + f 0 (a)(x − a) +
(x − a)2
2!
The cubic approximation to f (x) neat a is
f 00 (a)
f 000 (a)
f (x) ≈ f (a) + f 0 (a)(x − a) +
(x − a)2 +
(x − a)3
2!
3!
The approximations are called Taylor Polynomials about x = a. (You learn this in Cal II.)
1. Use differentials to approximate the following.
(a) cos(59o )
(b) sin(61o )
√
(c) 4 16.1
2. Find the linear approximation of f (x) =
√
√
x + 7 at a = 2 Use it to evaluate 9.06
3. Use Newton’s Method to find the solutions to x5 = 5x − 2 accurate to 4 decimal places.