Video Transcript
Find the linear approximation of
the function 𝑓 of 𝑥 equals 𝑥 sin of 𝑥 at 𝑥 equals two 𝜋.
Remember, if 𝑓 is differentiable
at 𝑥 equals 𝑎, then the equation for the tangent line approximation is given by 𝑙
of 𝑥 equals 𝑓 of 𝑎 plus 𝑓 prime of 𝑎 times 𝑥 minus 𝑎. In this example, we can let 𝑎 be
equal to two 𝜋. We’re going to need to evaluate 𝑓
of 𝑎 and 𝑓 prime of 𝑎. Let’s begin with 𝑓 of 𝑎. In this case, that’s 𝑓 of two
𝜋. So we substitute 𝑥 is equal to two
𝜋 into 𝑥 sin 𝑥. And we get two 𝜋 times sin of two
𝜋. We should know that sin of two 𝜋
is equal to zero. So 𝑓 of two 𝜋 is two 𝜋 times
zero which is just zero.
Now, 𝑓 prime of 𝑎 is going to
require a little more work. We’ll find the derivative of our
function. That’s the derivative of 𝑥 sin 𝑥
with respect to 𝑥, noticing that we have a function which is itself the product of
two differentiable functions. We’ll, therefore, use the product
rule. This says that, for two
differentiable functions 𝑢 and 𝑣, the derivative of their product is 𝑢 times d𝑣
by d𝑥 plus 𝑣 times d𝑢 by d𝑥. For our function, we’ll let 𝑢 be
equal to 𝑥 and 𝑣 be equal to sin 𝑥.
We’re going to need to
differentiate each of these with respect to 𝑥. d𝑢 by d𝑥 is one. And here, we recall the derivative
of sin 𝑥 with respect to 𝑥 is cos of 𝑥. And we substitute these into our
formula for the product rule. And we see that the derivative 𝑓
prime of 𝑥 is equal to 𝑥 times cos of 𝑥 plus sin 𝑥 times one. That’s 𝑥 cos 𝑥 plus sin 𝑥. To find 𝑓 prime of two 𝜋, we’ll
evaluate this when 𝑥 is equal to two 𝜋. That gives us two 𝜋 times cos of
two 𝜋 plus sin of two 𝜋. We already said that sin of two 𝜋
is zero. And cos of two 𝜋 is one. So 𝑓 prime of two 𝜋 is two 𝜋
times one plus zero which is simply two 𝜋.
Let’s substitute everything we now
have into the tangent line approximation formula. 𝑓 of 𝑎 is zero. 𝑓 prime of 𝑎 is two 𝜋. And 𝑥 minus 𝑎 is 𝑥 minus two
𝜋. We distribute our parentheses. And we see that the linear
approximation of our function 𝑓 of 𝑥 equals 𝑥 sin 𝑥 at 𝑥 equals two 𝜋 is two
𝜋𝑥 minus four 𝜋 squared.
