The derivative as the Rate of Change
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1) The area of a circle is given by
the equation A = π⋅
r2
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a) At what rate does the area of a circle
change as its radius changes?
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The derivative of A with
respect to r is A′, so
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A′ =
2π⋅
r
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b) If the radius is 3 how is the area
changing with respect to r?
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2 π⋅3 =
6π
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If the radius is 10?
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2 π⋅10
= 20π
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As you can see for larger radii, the area
changes at a higher rate for each change in r.
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NOTE: since the
phrase “with respect to” is heavily used in these
problems, it is usually abbreviated to the three letters
wrt.
2) What is the rate
of change of the volume of a cube wrt one of its sides?
Let the side of the cube be s. Then its volume
V = s3.
So dV/ds = 3s2
3)How does the area
of a square change wrt its diagonal?
Let the diagonal be d. We need the side s to
find the area. The
Pythagorean theorem relates the diagonal to the
side as follows:
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4) Find the values of x for which
the rate of change of y is zero for
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The rate of change is the derivative so we
need to find dy/dx. If we then set dy/dx to zero we can
then solve for x to find the answers.
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Next find the derivative of each
term.
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Now set this equation to zero and solve
for x.
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0 =
x2 + x − 6
0 = (x +
3)(x − 2)
0 = x + 3 or 0 = x
– 2
-3 = x 2 =
x
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The values of x that set the rate of
change to zero are -3, and
2.
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5) The
volume, V, of a
right circular cone is V =
1/3
r2
h. Find the
rate of change of its it height h with respect to its
radius r if its volume does not change.
We need to be careful here. As stated the
height and the radius are variable. The volume is
constant. If the base of the cone gets larger its height
must get smaller. Likewise if the base of the cone gets
smaller then its height must increase. The equation
dictates this. On the left V does not change. Looking on
the right, r and h are multiplied, so as one changes the
other must change in the opposite way to keep their
product constant.
The rate of change is wrt r so our
derivative is wrt r. Also, h depends on r so it acts as a
function and we need to use the product rule for
derivatives.
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The result tells us that as r increases h
must decrease twice as fast to keep the volume constant.
(The minus sign means decrease, and the 2 means
twice.)
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Higer Order Derivative applications coming soon!
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