1. ## Tangent Curve Questions

Couple of curve tangent questions. I have no idea what the 1st is asking.

2. Originally Posted by SportfreundeKeaneKent
Couple of curve tangent questions. I have no idea what the 1st is asking.

First Question:

Let curve 1 be: $\displaystyle 2x^2 + y^2 = 3 \Rightarrow y = \sqrt {3 - 2x^2}$
Let curve 2 be: $\displaystyle x = y^2 \Rightarrow y = \sqrt {x}$

Thus, the curves intersect where:
$\displaystyle \sqrt {3 - 2x^2} = \sqrt {x}$

$\displaystyle \Rightarrow 3 - 2x^2 = x$

$\displaystyle \Rightarrow 2x^2 + x - 3 = 0$

$\displaystyle \Rightarrow (2x + 3)(x - 1) = 0$

$\displaystyle \Rightarrow x = - \frac {3}{2} \mbox { or } x = 1$

$\displaystyle x = - \frac {3}{2}$ is extraneous, since $\displaystyle y$ has no solution for this $\displaystyle x$-value

when $\displaystyle x = 1$:

$\displaystyle y = \sqrt {1} = 1$

So $\displaystyle (1,1)$ is the ONLY point of intersection. Now let's find the slopes of the tangent lines of each curve at this point. If the slopes are perpendicular, that is, one is the negative inverse of the other, then we have that the curves are orthogonal.

For curve 1:

$\displaystyle 2x^2 + y^2 = 3$

$\displaystyle \Rightarrow 4x + 2y \frac {dy}{dx} = 0$

$\displaystyle \Rightarrow \frac {dy}{dx} = - \frac {2x}{y}$

at $\displaystyle (1,1)$, $\displaystyle \frac {dy}{dx} = -2$

For curve 2:
$\displaystyle x = y^2$

$\displaystyle \Rightarrow 1 = 2y \frac {dy}{dx}$

$\displaystyle \Rightarrow \frac {dy}{dx} = \frac {1}{2y}$

at $\displaystyle (1,1)$, $\displaystyle \frac {dy}{dx} = \frac {1}{2}$

Thus we see that the curves are orthogonal, since -2 is the negative reciprocal of $\displaystyle \frac {1}{2}$

Question 14:

$\displaystyle y = (x - 1)^2 (4 - x)$

By the product rule:

$\displaystyle \frac {dy}{dx} = 2(x - 1)(4 - x) - (x - 1)^2$

$\displaystyle \Rightarrow \frac {dy}{dx} = -3(x - 2)^2 + 3$ I skipped a lot of steps, i don't think you'll have a problem getting here.

This is a downward opening parabola, the derivative is maximum at it's vertex. That is, $\displaystyle y'$ is max at $\displaystyle (2,3)$

So we have that the tangent of greatest slope occurs at $\displaystyle x = 2$ and it's value is 3. I think you can take it from here and find the equation of the tangent line

3. Originally Posted by Jhevon
First Question:

$\displaystyle x = - \frac {3}{2}$ is extraneous, since $\displaystyle y$ has no solution for this $\displaystyle x$-value

when $\displaystyle x = 1$:

$\displaystyle y = \sqrt {1} = 1$

So $\displaystyle (1,1)$ is the ONLY point of intersection. Now let's find the slopes of the tangent lines of each curve at this point. If the slopes are perpendicular, that is, one is the negative inverse of the other, then we have that the curves are orthogonal.
Shouldn't (1,-1) also be a point of intersection since 1=y^2 and y= +/- 1 or am I plugging it into the wrong equation here or something?

4. Originally Posted by SportfreundeKeaneKent
Shouldn't (1,-1) also be a point of intersection since 1=y^2 and y= +/- 1 or am I plugging it into the wrong equation here or something?
you're absolutely right, my boy! (1,-1) is also a an intersecting point, my apologies. good looking out

5. ## the two curves

The two curves are perpendicular at the points of intersection.

6. Originally Posted by curvature
The two curves are perpendicular at the points of intersection.
scale on the x and y axes are different, which alters the apparent angle
of intersection

RonL

7. ## the largest slope

the largest slope occurs at the piont of inflection

8. ## visualize the problem

Originally Posted by CaptainBlack
scale on the x and y axes are different, which alters the apparent angle
of intersection

RonL
I just want to visualize the problem.

9. ## wrong

Originally Posted by Jhevon
First Question:

Let curve 1 be: $\displaystyle 2x^2 + y^2 = 3 \Rightarrow y = \sqrt {3 - 2x^2}$
Let curve 2 be: $\displaystyle x = y^2 \Rightarrow y = \sqrt {x}$

Thus, the curves intersect where:
$\displaystyle \sqrt {3 - 2x^2} = \sqrt {x}$

$\displaystyle \Rightarrow 3 - 2x^2 = x$

$\displaystyle \Rightarrow 2x^2 + x - 3 = 0$

$\displaystyle \Rightarrow (2x + 3)(x - 1) = 0$

$\displaystyle \Rightarrow x = - \frac {3}{2} \mbox { or } x = 1$

$\displaystyle x = - \frac {3}{2}$ is extraneous, since $\displaystyle y$ has no solution for this $\displaystyle x$-value

when $\displaystyle x = 1$:

$\displaystyle y = \sqrt {1} = 1$

So $\displaystyle (1,1)$ is the ONLY point of intersection. Now let's find the slopes of the tangent lines of each curve at this point. If the slopes are perpendicular, that is, one is the negative inverse of the other, then we have that the curves are orthogonal.

For curve 1:

$\displaystyle 2x^2 + y^2 = 3$

$\displaystyle \Rightarrow 4x + 2y \frac {dy}{dx} = 0$

$\displaystyle \Rightarrow \frac {dy}{dx} = - \frac {2x}{y}$

at $\displaystyle (1,1)$, $\displaystyle \frac {dy}{dx} = -2$

For curve 2:
$\displaystyle x = y^2$

$\displaystyle \Rightarrow 1 = 2y \frac {dy}{dx}$

$\displaystyle \Rightarrow \frac {dy}{dx} = \frac {1}{2y}$

at $\displaystyle (1,1)$, $\displaystyle \frac {dy}{dx} = \frac {1}{2}$

Thus we see that the curves are orthogonal, since -2 is the negative reciprocal of $\displaystyle \frac {1}{2}$

Question 14:

$\displaystyle y = (x - 1)^2 (4 - x)$

By the product rule:

$\displaystyle \frac {dy}{dx} = 2(x - 1)(4 - x) - (x - 1)^2$

$\displaystyle \Rightarrow \frac {dy}{dx} = -3(x - 2)^2 + 3$ I skipped a lot of steps, i don't think you'll have a problem getting here.

This is a downward opening parabola, the derivative is maximum at it's vertex. That is, $\displaystyle y'$ is max at $\displaystyle (2,3)$

So we have that the tangent of greatest slope occurs at $\displaystyle x = 2$ and it's value is 3. I think you can take it from here and find the equation of the tangent line
No, $\displaystyle y'$ is max at $\displaystyle (2,2).$