Page 80 - Calculus Workbook For Dummies

64 Part III: Differentiation

Solutions for Differentiation Basics

miles dp
a If you leave your home at time = 0, and go 60 in your car, what’s , the derivative of
hour dt
dp
your position with respect to time? The answer is = 60. A derivative is always a rate, and
dt
a rate is always a derivative (assuming we’re talking about instantaneous rates). So, if your
miles dp
speed, or rate, is 60 hour , the derivative, dt , is also 60. One way to think about a derivative
dp
like is that it tells you (in this case) how much your position (p) changes when the time (t)
dt
increases by one hour. A rate of 60 miles means that your position changes 60 miles each time
hour
the number of hours of your trip goes up by 1.
b Using the information from problem 1, write a function that gives your position as a function
of time. p t = 60 t or p 60= t, where t is in hours and p is in miles. If you plug 1 into t, your
^ h
position is 60 miles; plug 2 into t and your position is 120 miles. p = 60 t is a line, of course, in
the form y = mx + b (where b = 0). So the slope is 60 and the derivative is thus 60. And again
you see that a derivative is a slope and a rate.
1 2 23 85
c What’s the slope of the parabola y = - x + x - at the point (7, 9)? The slope is 3.
3 3 3
2
You can see that the line, y = 3 x - 12, is tangent to the parabola, y = - 1 x + 23 x - 85 , at the
3 3 3
point (7, 9). You know from y = mx + b that the slope of y = 3 x - 12 is 3. At the point (7, 9),
the parabola is exactly as steep as the line, so the derivative (that’s the slope) of the parabola
at (7, 9) is also 3.
Although the slope of the line stays constant, the slope of the parabola changes as you climb
up from (7, 9), getting less and less steep. Even if you go to the right just 0.001 to x = 7.001, the
slope will no longer be exactly 3.
2
d What’s the derivative of the parabola y = - x + 5 at the point (0, 5)? The answer is 0.
2
The point (0, 5) is the very top of the parabola, y = - x + 5. At the top, the parabola is neither
going up nor down — just like you’re neither going up nor down when you’re walking on top
of a hill. The top of the parabola is flat or level in this sense, and thus the slope and derivative
both equal zero.
The fact that the derivative is zero at the top of a hill — and at the bottom of a valley — is a
critically important point which we’ll return to time and time again.

2
e With your graphing calculator, graph both the line y = - 4 x + 9 and the parabola y = - x .
5
You’ll see that they’re tangent at the point (2, 1).
5
2
a. What is the derivative of y = - x when x = 2? The answer is –4. The derivative of a curve
tells you its slope or steepness. Because the line and the parabola are equally steep at (2, 1),
and because you know the slope of the line is –4, the slope of the parabola at (2, 1) is also –4
and so is its derivative.
b. On the parabola, how fast is y changing compared to x when x = 2? It’s decreasing 4 times as
fast as x increases. A derivative is a rate as well as a slope. Because the derivative of the
parabola is –4 at (2, 1), that tells you that y is changing 4 times as fast as x, but because the 4
is negative, y decreases 4 times as fast as x increases. This is the rate of y compared to x only
for the one instant at (2, 1) — and thus it’s called an instantaneous rate. A split second later —
say at x = 2.000001 — y will be decreasing a bit faster.