Math 201, Section 3

September 22, 2004

Walter Stromquist

Linear Approximations I

If is a vector-valued function, and is a number,

then the linear approximation to r at the point is given by

,

defined for every value of t.

Example 1: What is the linear approximation to the function

at the point ?

Solution: In this case, t₀ = 1. First compute the derivative of the function,

,

and evaluate it at t₀ = 1:

.

Then just substitute the known values of t₀, r(1) and into equation :

.

What does it mean for to be an approximation to r(t)?

Informally: When t is close to t₀, then is close to r(t).

Formally: When t approaches t₀, approaches r(t). What’s more, the difference

approaches zero even faster than (t-t₀) does. Specifically:

.

(You can prove that if you substitute the definition of into that last equation and keep in mind the definition of the derivative.)

Why is it called a linear approximation ?

Because in general, a linear function from to is any function of the form

where a and b are any constant vectors. (They’re “constant” in the sense that they don’t depend on t.)

The graph of a linear function is a line. (That is, unless b is the zero vector. Then the graph is just a single point but we call l a linear function anyway.)

Equation isn’t exactly in the form of , but we can rearrange it easily:

.

The expressions in the square brackets are vectors that don’t depend on t, so this is clearly a linear function.

If you do this with the result of Example 1, you get

.

Usually there is no need to make this transformation. We only derived in order to show that l satisfies the definition of a linear function. Use whichever version you like.

Why did we write the last term in that way? It’s just scalar multiplication. Don’t we usually write the number before the vector?

Yes. The equation might be a little clearer if we wrote it like this:

,

But I’m trying to establish a pattern here. We’re going to see linear approximations again, in situations where the original form of equation makes more sense.

What if the derivative doesn’t exist?

Then there is no linear approximation to r at r(t₀).

What if the derivative exists, but is equal to zero?

Then equation still applies, and so does equation . The linear approximation is one of those silly one-point functions.

Example 2: What is the linear approximation to the function

at the point ?

Solution: This function is a helix-like curve, except that the spirals keep getting wider as they move above the xy-plane. The derivative is:

which, evaluated at , gives

so the linear approximation is

.

Example 2 continued: What is ? (Note: )

Solution: Since the argument is so close to t₀, we can get a pretty good answer by substituting into equation . We get

Compare this to the result we get directly from the original function:

Example 3: What is the linear approximation to the function

at the point ?

Solution: This based on Example 2, page 894 in the text. It is just like the examples above, except that you aren’t given t₀.

But, obviously, if the given point is r(t₀), then t₀=p/2. So you have all you need. Compute:

.

We can compare this to the text version of the example. The text only asks for the equation of a tangent line to the curve, and it’s answer is (rewritten as a vector equation)

.

The difference is that the text only wants a tangent line to the curve; it doesn’t care whether the parameter values match. We are asking for a linear approximation to the function, which is something slightly better.

What’s going to be on the quiz?

A problem just like Example 1 or the first part of Example 2.

More practice: Try problems 27 and 28 in the text, page 897, but find a linear approximation at the given point. (Your answer to 27 won’t quite agree with the answer in the back of the book.)

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