## Calculus question

Forum for the GRE subject test in mathematics.
CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm

### Calculus question

If f(x)=\int_0^x{x^2sin(t^2)dt}, then what is f'(x)?

CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm
I have not solved this. Here is a simpler one, but with a good idea.

Let a curve be defined parametrically by

x=\int_1^t{\frac{cos u}{u}du},
y=\int_1^t{\frac{sin u}{u}du}.

Find the arc length from the origin to the nearest point on the curve with a vertical tangent.

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
If f(x)=\int_0^x{x^2sin(t^2)dt}, then what is f'(x)?
Can we use the formula
f(x)=\int_a(x)^b(x){g(t)dt}------>f'(x)=g[b(x)]b'(x)-g[a(x)]a'(x) Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
I have not solved this. Here is a simpler one, but with a good idea.

Let a curve be defined parametrically by

x=\int_1^t{\frac{cos u}{u}du},
y=\int_1^t{\frac{sin u}{u}du}.

Find the arc length from the origin to the nearest point on the curve with a vertical tangent.
the arc length l=int_0^a(sqrt[x'(t)^2+[y'(t)^2])dt
if x=x(t)
y=y(t) then y'(x)=dy/dx=y'(t)/x'(t)----------> from this equation solve for a
and calculate the integral CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm
Can we use the formula
f(x)=\int_a(x)^b(x){g(t)dt}------>f'(x)=g[b(x)]b'(x)-g[a(x)]a'(x)
Not when there is a function of x in the integrand.

CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm
the arc length l=int_0^a(sqrt[x'(t)^2+[y'(t)^2])dt
if x=x(t)
y=y(t) then y'(x)=dy/dx=y'(t)/x'(t)----------> from this equation solve for a
and calculate the integral
Very close, but one detail needs to be changed.

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
If f(x)=\int_0^x{x^2sin(t^2)dt}, then what is f'(x)?
Use the Leibnitz's rule :http://en.wikipedia.org/wiki/Leibniz_integral_rule

CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm
Still must finish. Go ahead & do it (you will have to do it twice) then tell me how you resolve the integral with only t in it