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?


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.

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