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Math 2413
Notes 3.1
Section 3.1 – Related Rates
Recall:
The Derivative as a Rate of Change
Given y = f (x), the slope of the secant line
f (b)  f (a )
ba
gives the average rate of change of y with
respect to x over the interval [a,b].
On the other hand, the slope of the tangent line, f '
(a) , gives the instantaneous rate of change
in y with respect to x at a .
The instantaneous rate of change is the limit
of the average rates of change as the interval
width approaches 0.
Examples of applications of rates of change.
Example 1: Let A denote the area inside a circle. The area depends on the radius of the circle denoted by r . If
the radius changes, then the area changes. How fast is the area changing at the instant when r = 2 or r = 3cm?
r =1
r =2
r=3
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Math 2413
Notes 3.1
Related Rates
Many interesting problems occur when a quantity changes with respect to time. Population, velocity, inventory,
traffic flow, heights of plants, etc. can be modeled as functions of time. We will now look at examples where
the independent variable is time.
Must know the formulas of:

Areas of squares, rectangles, triangles, and trapezoids.

Volumes of cubes, spheres, cylinders, and cones.

Surface area of cubes, cylinders and spheres.
Example 2:
1. Give the rate of change of the surface area of a sphere with respect to its radius r.
2. Give the rate of change of the volume of a sphere with respect to
a) its radius r
b) time.
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Math 2413
Notes 3.1
3. Suppose the height h of a right circular cone is always twice the radius r. Give the rate of change of the
volume of this cone with respect to h, and evaluate this rate when h = 2.
Example 3: Translate the following sentences using proper calculus notation.
1. The area of a circle is increasing at a rate of six square inches per minute.
2. The volume of a cone is decreasing at a rate of two cubic feet per second.
3. The population of Bigtown is growing at a rate of 3 people per day.
4. The height of a tree is increasing at a rate of ½ foot per year.
5. The water level in my fish tank is decreasing at a rate of 2 inches per hour.
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Math 2413
Notes 3.1
Example 4: Suppose that a spherical snowball is melting in a manner that the shape remains a sphere. Given
that the radius is decreasing at a rate of 1.5 inch per minute, at what rate is the volume changing when the radius
is 12 inches?
Example 5: A water tank has the shape of a cone with height 5 meters and radius 2 meters. If water is pumped
into the tank at a rate of 3 cubic meters per hour, how fast is the water level rising at the instant the water
reaches a depth of 2 meters?
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Math 2413
Notes 3.1
Example 6: A rocket is fired from the ground straight up and a camera located 500 feet away is following the
rocket. The rocket is rising at the rate of 250 feet per second when it is 300 feet up.
a) How fast is the camera-to-rocket distance changing at this instant?
b) At this instant, how fast must the camera elevation angle change to keep the rocket in sight?
Example 7: A particle is moving along the curve y  x 2  x 3 in such a way that the y − coordinate is
decreasing at a rate of 4 units per minute. At what rate is the x – coordinate changing as the particle passes
through the point (-1,0)?
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Math 2413
Notes 3.1
Velocity and Acceleration
Velocity
Let s(t) be the position of the particle at time t measured relative to some reference point (where s = 0 ). If the
position function s(t) is differentiable, then the derivative s' (t) gives the rate of change of the position function
at time t . This rate is called the velocity at time t and is denoted as v(t). In symbols, v(t) = s' (t).
Acceleration
Acceleration is defined to be the rate of change of velocity per unit time. If the velocity function
v(t) is differentiable, then its derivative gives the acceleration function; a(t) = v' (t).
Given the position function s(t), the instantaneous rate of change of an object is the velocity of the object:
s ' (t )  v(t )
The instantaneous rate of change of the velocity is the acceleration of the object:
a (t )  v ' (t )  s ' ' (t )
Other notation:
v
ds
dt
a
and
dv d 2 s

dt dt 2
Summary:
Speed is the absolute value of velocity.
Velocity gives direction and magnitude.
Speed gives magnitude.
Speed at time t = | v(t) |
We can look at the sign of each (velocity and acceleration) to determine if an object is speeding up or slowing
down and the direction it is going. If the signs are the same, the object is speeding up.
Newton’s Second Law relates acceleration, mass and force: F = ma.
So, whenever there is acceleration, there’s force and vice versa.
If the signs are different, the object is slowing down.
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Math 2413
Notes 3.1
Example 8: An object moves along a coordinate line, its position at each t ≥ 0 the time given by x(t). Find the
position, velocity, and acceleration and speed at time t0.
x(t) = 5t – t3 ; t0 = 3.
Example 9:
1. A particle is moving along a horizontal coordinate line according to the formula s = t 3 – 6t2.
a/ Find v(t).
b/ When is the particle moving to the left?
c/ When is the acceleration negative?
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Math 2413
Notes 3.1
d/ When is the particle speeding up?
e/ When is the particle slowing down?
Free Fall Formulas
The height of an object in free fall is given by
h(t )  16t 2  v0t  h0
(distance in feet)
h(t )  4.9t 2  v0t  h0
(distance in meters),
Example 10: A stone is thrown straight up from a height of 8 feet with an initial velocity of 62
feet per second.
a) When does the stone reach a height of 50feet?
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Math 2413
Notes 3.1
b) What is the speed of the stone when it reaches 50 feet?
c) What is its maximum height?
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