We're asked to find the linear approximation
for f of x equals sine 3x at x equals zero
and then use the linear approximation to estimate f of 0.1.
The linear approximation for f of x at x equals zero
would be the equation of the tangent line
to f of x at the point where x equals zero,
which means we to determine the point of tangency
and the slope of the tangent line
to determine the linear approximation
for the given function.
Let's first determine the point on the function
where x equals zero.
If x equals zero,
then the y-coordinate is equal to f of zero,
which is the sine of 3 times zero,
which equals the sine of zero, which is zero.
So now we know, when the x-coordinate is zero,
so is the y-coordinate.
And therefore we're looking for the equation
of the tangent line at the point zero comma zero.
Next, we need to determine the slope of the tangent line
at the point zero comma zero.
And the slope of the tangent line
is equal to f prime of zero.
So given f of x equals the sine of 3x,
we need to find the derivative function,
which notice how does require the chain rule,
where the inner function u is equal to 3x.
F prime of x
is equal to the derivative of sine u with respect to x,
which is cosine u times u prime,
or in our case, the cosine of 3x times derivative of 3x
with respect to x, which is 3,
giving us f prime of x equals 3 times the cosine of 3x.
So now we can find the slope of the tangent line
by determining f prime of zero,
which is 3 times the cosine of 3 times zero,
which is 3 times the cosine of zero.
The cosine of zero is 1. 3 times 1 is 3.
So now we know the slope of the tangent line is 3.
So now that we have the slope of tangent line
and the point of tangency,
we can determine the linear approximation
for f of x at x equals zero.
Let's go ahead and use point-slope form of a line,
which is y minus y-one
equals m times the quantity x minus x-one,
where x-one comma y-one is the point and m is the slope.
This gives us y minus zero
equals 3 times the quantity x minus zero,
which simplifies nicely to y equals 3x.
So if we call the linear approximation the function L of x,
we now know L of x equals 3x.
And now we're asked to use this linear approximation
to estimate f of 0.1.
So the idea here is that f of 0.1,
which is equal to the sine of 3 times 0.1,
is going to be approximately equal to L of 0.1,
which is equal to 3 times 0.1, or just 0.3.
So using a linear approximation, again, we have f of 0.1,
which is exactly equal to the sine of 0.3,
is approximately equal to 0.3.
Let's take a look at this graphically.
In blue, we have the graph of the given function
f of x equals sine 3x.
The linear approximation at x equals zero
is this red tangent line,
and to approximate f of 0.1,
we use a linear approximation
to get the function value of 0.3,
which we can see graphically,
as well as from the calculation below,
that it is very close to the actual function value,
f of 0.1.
I hope you found this helpful.