# Thread: Integration and Differentiation Problem

1. ## Integration and Differentiation Problem

I took a math exam and want to see if I got these questions right. These are the ones I can remember.

3.) Integral of (sin x cos x) from x = -pi to x = pi.

We weren't allowed to ask questions so I didn't know if they wanted net area or all the area counted as positive, even the negative area so I put answer = 0.

6a.) Find dy/dx for the following equation of an hyperbola x^2 - y^2 = 9.

I got the answer dy/dx = -x/-y

6b.) What is the equation of the tangent line to the hyperbola at (3,0)

I put no such tangent line exists.

2. ## re: Integration and Differentiation Problem

Originally Posted by Rstewart
3.) Integral of (sin x cos x) from x = -pi to x = pi.

We weren't allowed to ask questions so I didn't know if they wanted net area or all the area counted as positive, even the negative area so I put answer = 0.
\displaystyle \begin{align*} f(x) &= \sin{(x)}\cos{(x)} \\ f(-x) &= \sin{(-x)}\cos{(-x)} \\ f(-x) &= -\sin{(x)}\cos{(x)} \\ f(-x) &= -f(x) \end{align*}

This is an odd function. You should know that \displaystyle \begin{align*} \int_{-a}^a{f(x)\,dx} = 0 \end{align*} if \displaystyle \begin{align*} f(x) \end{align*} is an odd function. So you are correct.

You could also have solved the integral using Brute Force.

3. ## re: Integration and Differentiation Problem

Originally Posted by Rstewart
6a.) Find dy/dx for the following equation of an hyperbola x^2 - y^2 = 9.

I got the answer dy/dx = -x/-y

6b.) What is the equation of the tangent line to the hyperbola at (3,0)

I put no such tangent line exists.
I agree with the calculation of your derivative, though I would simplify it to x/y.

Anyway, for part b, what do you know about the gradients of vertical lines?

4. ## re: Integration and Differentiation Problem

Originally Posted by Prove It
I agree with the calculation of your derivative, though I would simplify it to x/y.

Anyway, for part b, what do you know about the gradients of vertical lines?
>.< So that means it would be a vertical tangent and I got that one wrong?

5. ## re: Integration and Differentiation Problem

Originally Posted by Rstewart
>.< So that means it would be a vertical tangent and I got that one wrong?
Yes, the equation of the tangent would have been \displaystyle \begin{align*} x = 3 \end{align*}.