We are given f double prime of x
equals negative 2 sine x, f prime of zero equals 5,
and f of zero equals negative 4.
We're asked to determine f of x
and the exact value of f of pi divided by 3.
When we determine an antiderivative
or evaluate an indefinite integral,
we can think of this as performing the opposite operation
of differentiation.
If you look at our notes below,
the indefinite integral of f of x with respect to x
is equal to big F of x plus C,
where big F of x plus C is the family of functions
whose derivative is the given integrand function of f of x.
Which means if we find the antiderivative
of the second derivative function,
we can recover the family of first derivative functions
whose derivative is f double prime.
And then because we have this initial condition,
f prime of zero equals 5,
we can determine the exact first derivative function
by determining the constant of integration.
So let's begin by integrating the second derivative function
of negative 2 sine x with respect to x,
which is equal to negative 2 times negative cosine x plus C,
which simplifies to 2 cosine x plus C.
So again, because we found the antiderivative
of the second derivative function,
we know the family of first derivative functions
must be in the form of f prime of x
equals 2 cosine x plus C.
But again, now we can refer to the initial condition
to determine the specific first derivative function,
because we know f prime of zero equals 5.
So if f prime of zero is equal to 5,
if we substitute zero x in the first derivative function,
the derivative function value must be equal to 5.
This gives us the equation 2 times cosine of zero
plus C equals 5.
Cosine zero is equal to 1,
giving us the equation 2 plus C equals 5.
Subtracting 2 on both sides, we have C equals 3.
So now we know the specific first derivative function
is f prime of x equals 2 cosine x plus 3.
And now that we have the specific first derivative function,
we can find the antiderivative of this function
to help us term f of x.
And now we'll evaluate the indefinite integral
of 2 cosine x plus 3 with respect to x,
which is equal to 2 sine x plus 3x plus C.
This is the family of functions
whose derivative is 2 cosine x plus 3,
which indicates that f of x must be in the form
of 2 sine x plus 3x plus C.
And again, because we are given f of zero equals negative 4,
we can determine the specific function f of x
by determining the value of C.
If f of zero is equal to negative 4,
if we substitute zero for x in f of x,
the function value must be negative 4,
which gives us the equation
2 times the sine of zero plus 3 times zero plus C
equals negative 4.
Simplifying on the left, sine zero is zero,
3 times zero is zero.
Our solution is C equals negative 4,
which means the specific function f of x,
in our case, is f of x equals 2 sine plus 3x minus 4.
So going back to the question,
we were asked to determine f of x, which we now have.
And now we're asked to determine the exact value
of f of pi divided by 3.
So f of pi divide by 3
is equal to 2 times the sine of pi divided by 3
plus 3 times pi divided by 3 minus 4.
For a quick review, if we sketch
a 30-60-90 reference triangle,
pi divided by 3 would be this angle here.
We can label the short leg 1, the hypotenuse 2,
then long leg square root 3.
Sine of pi divided by 3 is equal to the ratio
of the opposite side to the hypotenuse,
or square root 3 divided by 2,
which gives us 2 over 1 times square root 3 divided by 2
plus 3 over 1 times pi divided by 3
minus 4.
Simplifying, 2 divided by 2 simplifies to 1.
3 divided by 3 simplifies the 1.
The exact value of f of pi divided by 3
is square root 3
plus pi
minus 4.
I hope you found this helpful.