# Derivatives, slopes of parallel lines and Related Rates (High School)

I'm a bit confused about some derivative and related rates stuff. I guess it's more like I'm not sure if I've done them correctly:

1.) The diameter of a tree was 12 in. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase?

So it's given that $\frac{dC}{dt} = 2$, and the value sought would be $\frac{dD}{dt}$. The two relevant formulas would presumably be the area and circumference of a circle (both expressed in terms of diameter rather than radius), but it seems to me that the problem can be done with just the Circumference formula, since I'm looking for $\frac{dD}{dt}$ and I have

$$C = 2(\pi)r = (\pi)D$$

So then all I'd have to do is differentiate with respect to time

$$\frac{dC}{dt} = \frac{dD}{dt}\pi$$

and substituting the given $\frac{dC}{dt} = 2$, I'd have $2 = \frac{dD}{dt}\pi$, and ultimately

$$\frac{dD}{dt} = \frac{\pi}{2}$$

The thing is, it seems "too easy" this way. Is this correct? Or am I supposed to be using the Area formula for something? Or am I way off?

2.) Find all points (x,y) on the graph of $f(x) = 2x^{2} - 3x$ with tangent lines parallel to the line $y = 13x +5$.

Of course, two lines are parallel if they have the same slope, and since the second equation is so simple in form, you can immediately tell that it has slope $m = 13$. Then considering the definition of a derivative ("the slope of a line tangent to a curve at a given point"), the "points" with tangent lines parallel to a given line should satisfy ${f}' = m$, with ${f}' = 4x - 3$ in this case, such that $x = 4$ and $y = f(4) = 20$.

That's just it, though. The "solution" in this case is one single point, (4, 20), but the question asks for "pointS". Is there some other point I'm missing?

## 1 Answer

For the first one it is correct, Area formula is not needed unless looking for change in area, dA/dt. For the second, y-y1=m(x-x1) -> y-(20)=13(x-4) -> y=13x-32. So when graphed f(x)=2(x^2)-3x is curved and touches the straight line,y=13x-32, at one point. Both are correct.