2.3 Velocity, Acceleration and Second Derivatives
GOAL: Explore the use of derivatives to analyze the motion of objects travelling in a straight line.
RECALL:
DEFINITION
RELATIONSHIP
DISPLACEMENT
𝒔(𝒕)
the distance and direction an object
has moved from an origin over a
period of time.
VELOCITY
𝒗(𝒕)
the rate of change of displacement
of an object with respect to time
ACCELERATION
𝒂(𝒕)
the rate of change of velocity with
respect to time
Example: The graph shows the position function of an object. Complete the following chart.
Interval
Description
of Slope of
𝒔(𝒕)
Velocity
𝒗(𝒕)
Acceleration
𝒂(𝒕)
(𝑨, 𝑩)
(𝑩, 𝑪)
(𝑪, 𝑫)
(𝑫, 𝑬)
RECALL:
Speed is a scalar quantity. It describes the magnitude of motion, but it does not describe direction.
Velocity is a vector quantity and has both magnitude and direction. The answer in a velocity problem will
either be a negative or a positive value. The original position is considered the origin. One direction from
the origin is considered positive and the opposite direction from the origin is considered negative.
Below is a graphical representation of:
• the position function, 𝑠(𝑡),
• the velocity function, 𝑠’(𝑡) or 𝑣(𝑡)
• the acceleration function, 𝑠’’(𝑡) or 𝑣’(𝑡) or 𝑎(𝑡)
On the graph of position, circle the zones where the object is speeding up and where the object is slowing
down.
Now consider the graph of acceleration, does a positive acceleration always correspond to speeding up?
Similarly, does a negative acceleration always correspond to slowing down?
Speeding Up versus Slowing Down
There are two possible cases for speeding up:
Case 1
An object is moving in a positive direction and the velocity becomes more positive. In this case 𝑣(𝑡) > 0
(moving in a positive direction) and 𝑎(𝑡) > 0 (the velocity is becoming more positive). Note that
𝑣(𝑡) × 𝑎(𝑡) > 0
Case 2
An object is moving in a negative direction and the velocity becomes more negative. In this case, 𝑣(𝑡) < 0
(moving in a negative direction) and 𝑎(𝑡) < 0 (velocity is becoming more negative). Note that
again, 𝑣(𝑡) × 𝑎(𝑡) > 0
An object is speeding up when 𝒗(𝒕) × 𝒂(𝒕) > 𝟎
Velocity and Acceleration are in the _____________ direction.
There are two possible cases for slowing down:
Case 1
An object moving in a positive direction, 𝑣(𝑡) > 0, and the velocity becomes less positive, 𝑎(𝑡) < 0. Note
that 𝑣(𝑡) × 𝑎(𝑡) < 0.
Case 2
An object moving in a negative direction, 𝑣(𝑡) < 0 and the velocity becomes less negative, 𝑎(𝑡) > 0. Note
that 𝑣(𝑡) × 𝑎(𝑡) < 0.
An object is slowing down when 𝒗(𝒕) × 𝒂(𝒕) < 𝟎
Velocity and Acceleration are in the ___________________________ direction.
Example: Find the velocity and the acceleration of the displacement function 𝑠(𝑡) = −5𝑡 ! + 2𝑡 " − 7𝑡
when 𝑡 = 3. State whether the object is speeding up or slowing down at 𝑡 = 3.
Example: The position function of an object moving along a straight line is represented by the function
𝑠(𝑡) = 2𝑡 ! – 3𝑡 " – 36𝑡 + 6, where 𝑡 is in seconds and 𝑠 is in metres.
a) What is the position of the object after 2 𝑠 and after 5 𝑠?
b) What is the velocity of the object after 2 𝑠 and after 5 𝑠?
c) When is the object stopped? What is its position at this time?
d) When is the object moving in a positive direction?
e) Determine the total distance travelled by the object during the first 10 𝑠.
2.6 Rate of Change Problems
GOAL: Apply derivatives to solve problems involving rates of change in the social and physical sciences.
Rates of Change in Business and Economics
Business Functions
• The demand function or price function, 𝑝(𝑥), is the price per unit that the marketplace is willing to
pay for a given product or service at a production level of 𝑥 units.
• The revenue function is _________________________ where 𝑥 is the number of units of a product or
service sold at a price per unit of 𝑝(𝑥).
• The cost function, 𝐶(𝑥), is the total cost of producing 𝑥 units of a product or service.
• The profit function, ___________________________ is the profit from the sale of 𝑥 units of a product or
service.
Derivatives of Business Functions
Economists use the word marginal to indicate the derivative of a business function.
•
$%
𝐶 # (𝑥) or $& is the marginal cost function and refers to the _________________ of ___________________ with
respect to _________________________________________.
•
$'
𝑅′(𝑥) or $& is the marginal _______________ function and refers to the instantaneous rate of change of
____________________________ with respect to ___________________________________.
•
$(
𝑃′(𝑥) or $& is the marginal _________ function and refers to the ____________ of ________________ with
respect to the ________________________________.
Example: A company sells 1500 movie DVDs per month at $10 each. Market research has shown that
sales will decrease by 125 DVDs per month for each $0.25 increase in price.
a) Determine the demand, or price, function.
b) Determine the marginal revenue when sales are 1000 DVDs per month.
c) The cost of producing 𝑥 DVDs is 𝐶(𝑥) = −0.004𝑥 " + 9.2𝑥 + 5000. Determine the profit and marginal
profit from the monthly sales of 1000 DVDs.
Example: Kinetic energy, 𝐾, is the energy due to motion. When an object is moving, its kinetic energy is
determined by the formula 𝐾(𝑣) = 0.5𝑚𝑣 " , where 𝐾 is in joules; 𝑚 is the mass of the object, in kilograms,
and 𝑣 is the velocity of the object, in metres per second. Suppose a ball with a mass of 350 𝑔 is thrown
vertically upward with an initial velocity of 40 𝑚/𝑠. Its velocity function is 𝑣(𝑡) = 40 − 9.8𝑡, where 𝑡 is
time, in seconds.
a. Express the kinetic energy of the ball as a function of time.
b. Determine the rate of change of the kinetic energy of the ball at 3𝑠.
DERIVATIVES AND LINEAR DENSITY
𝑚𝑎𝑠𝑠
𝐿𝑖𝑛𝑒𝑎𝑟 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑙𝑒𝑛𝑔𝑡ℎ
Used when examining objects of a constant shape.
• Objects made of homogeneous materials will have a constant linear density.
• Objects made of non-homogeneous materials will have a density that changes along the object’s length.
Let 𝑓(𝑥) = 𝑚𝑎𝑠𝑠, in kilograms, of the first 𝑥 metres of the object. Then, the average linear density of the
object between 𝑥 = 𝑥) and 𝑥 = 𝑥" is defined as:
*(& )-*(& )
𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = &! -& "
.
!
"
The corresponding derivative function 𝑓’(𝑥) is the
Example: The mass, in kilograms, of the first 𝑥 metres of a wire can be modelled by the function
𝑓(𝑥) = √3𝑥 + 1.
a. Determine the average linear density of the part of wire from 𝑥 = 5 to 𝑥 = 8.
b. Determine the linear density at 𝑥 = 5 and 𝑥 = 8. Compare the densities at the two points. What
do these values confirm about the wire?
3.6 Optimization Problems
GOAL: Examine a variety of real-life problems in which it is necessary to find the best, or optimal value.
An optimization problem requires you to find a maximum or a minimum value.
Steps For Solving Optimization Problems:
1. Draw a diagram and identify what the question is asking
2. Identify the quantity to be ___________________________ and write the equation
3. Define the independent variable. Express all other _____________________ in terms of the independent
variable. Identify and state any _______________________ on the independent variable.
4. Define the function in terms of the independent variable.
5. ___________________________ the function.
6. Determine and classify the _______________________ points. Analyze the results
• Method 1: Increasing/Decreasing Chart
• Method 2: Examine Test Points to the left and right
• Method 3: Use the Second Derivative to Test Concavity
o If 𝑓 # (𝑎) = 0 and 𝑓 ## (𝑎) > 0, then (𝑎, 𝑓(𝑎)) is a local _______________________
o If 𝑓 # (𝑎) = 0 and 𝑓 ## (𝑎) < 0, then (𝑎, 𝑓(𝑎)) is a local _______________________
Example: A cattle rancher has purchased five 𝟏𝟓𝟎𝒎 rolls of wire fencing to build a rectangular corral. She
will use all the fencing. What dimensions will produce the greatest possible area?
Example: A cylindrical storage tank with a capacity of 𝟏𝟎𝟎𝟎 𝒎𝟑 is to be constructed in a warehouse that
is 𝟏𝟐 𝒎 by 𝟏𝟓 𝒎 and has a height of 𝟏𝟏 𝒎. The specifications call for the base to be made of sheet steel,
which costs $𝟏𝟎𝟎/𝒎𝟐 , the top of sheet steel, which costs $𝟓𝟎/𝒎𝟐 , and the wall of sheet steel which costs
$𝟖𝟎/𝒎𝟐 . Find the proportions of the tank to minimize the cost of construction.
Example: A carpenter is building an open box with a square base for holding firewood. The box must
have a surface area of 𝟖 𝒎𝟐 . What dimensions will yield the maximum volume?
OPTIMIZATION PROBLEMS
1. 200 m of fencing was purchased to create a rectangular dog park. Determine the dimensions of the
park that would give the maximum area.
2. A school principal wants to put up a rectangular fence next to the
school. The principal wants the area within the fence to be 300
What is the minimum amount of fencing the principal needs to
purchase? (Fencing is only needed on three sides.)
m2 .
3. Squares are cut out of the corners of a piece of sheet metal which is then folded to make a wading
pool. If the piece of sheet metal is 14 m by 8 m, what are the dimensions of the wading pool with the
largest volume that can be made from the sheet metal?
4. A house owner has already purchased 10 m of fencing. He wants to build a rectangular play area for
his child that is 400 m2. What is the minimum amount of fence that he needs to buy to build the play
area?
5. Wire mesh enclosures for two batting cages are to be built side
side at a school. If each cage needs 240 m3 of space, what is the
minimum amount of wire mesh needed, where 3 £ x £ 5 ?
(Wire mesh is not needed on the bottom of the cage.)
by
6. A packaging company wants to create a new can for their soup. The volume of the can must be 200
cm3. What are the dimensions of the can that will minimize surface area?
ANSWERS
1. 50 m x 50 m
2. 49 m
3. 1.6 m x 10.8 m x 4.8 m
4. 70 m
5. 340 m2
6. r = 3.2 cm, h = 6.2 cm
4.3 Differentiation Rules for Sinusoidal Functions
GOAL: Apply the rules of derivatives to the sine and cosine functions.
The Chain Rule, Power of a Function Rule and Product Rule that we learned previously all apply to
sinusoidal functions.
Chain Rule
Power of a Function Rule
Product Rule
Example: Find the derivative with respect to x for each function.
a) 𝑦 = sin 5𝑥
b) 𝑦 = cos 𝜋𝑥
c) 𝑦 = 6 sin 3𝑥 − sin 𝑥
)
e) 𝑦 = 012 &
d) 𝑦 = sin! 𝑥 − 2 cos 𝑥
f) 𝑦 = cos(−𝜋𝑥 + 3)
Differentiate with respect to 𝑡.
a) 𝑦 = sin" (𝑡 − 3)
b) 𝑓(𝑡) = sin (3𝑡 " ) + cos(4𝑡)
c) 𝑓(𝑡) = sin(cos " 𝑡)
Example: Find the slope of the tangent line to 𝑦 = 2 sin 𝑥 cos 𝑥 at 𝑥 = 𝜋.
5.4 Differentiation Rules for Exponential Functions
GOAL: Apply the rules of differentiation to exponential functions.
Example: Find the derivative of each function.
a) 𝑦 = 𝑒 !&
c) 𝑦 = 𝑒 &
e) 𝑦 =
#
4$
&
g) 𝑔(𝑥) = 5& 𝑒 &
b) 𝑦 = 𝑒 3-!&
d) 𝑦 = 𝑥 " 𝑒 "&
f) 𝑓(𝑥) = (1 + 5𝑒 !& )"
4.4 Applications of Sinusoidal Functions and Their Derivatives
GOAL: Apply derivatives to solve problems involving sinusoidal functions.
Example: An AC-DC coupled circuit produces a voltage described by the function,
𝑉(𝑡) = 5 sin 𝑡 + 8, where 𝑉 is the voltage, in volts and 𝑡 is the time, in seconds.
a) Determine the first derivative of the voltage function.
b) Find the maximum and minimum voltages. At what time do these values occur?
Example: A simple pendulum has a length of 20𝑚 and a maximum horizontal displacement of 8𝑚.
a) Determine a function that gives the horizontal position as a function of time.
b) Determine the function that gives the velocity as a function of time.
c) Find the maximum velocity.
Example: A geometric figure is bouncing on a spring that is hanging from the ceiling. The figure’s distance
from the ceiling, 𝑑, varies sinusoidally with time, 𝑡. The figure starts at its closest point to the ceiling, 𝑑 =
0.5𝑚 and makes a complete cycle every second. The figure’s farthest point from the ceiling is 1.5𝑚.
a) Sketch a graph that shows the figures height from the ceiling over time for 1 cycle. Determine the
function for the height, ℎ, in terms of time, 𝑡.
b) How far from the ceiling is the figure when 𝑡 = 2?
c) Determine the function for the velocity of the figure in terms of time.
d) How fast is the figure moving when 𝑡 = 2?
e) What is the fastest speed of the figure?
f) When is its velocity equal to zero?
5.5 Making Connections: Exponential Models
GOAL: Apply derivatives to solve problems involving exponential functions.
Exponential functions and their derivatives are important modeling tools for a variety of fields of study
such as: nuclear engineering, mechanical engineering, electronics, biology, and environmental science.
Example 1: Exponential Decay
The amount of a radioactive material as a function of time is given by the standard function
𝑁(𝑡) = 𝑁5 𝑒 -67 where:
𝑁 = the number of radioactive nuclei at time t
𝑁5 = the initial number of radioactive nuclei
𝜆 = the disintegration constant
A radioactive isotope of gold, Au-198 is used in the diagnosis and treatment of liver disease. Suppose that
a 6.0 𝑚𝑔 sample of Au-198 is injected into a liver and that this sample decays to 4.6 𝑚𝑔 after 1 day.
Assume the amount of Au-198 remaining after 𝑡 days is given by 𝑁(𝑡) = 𝑁5 𝑒 -67 .
a) Determine the disintegration constant for Au-198.
b) Determine the half-life of Au-198
c) Write the equation that finds the amount of Au-198 remaining as a function of time, in terms of its
half-life.
d) How fast is the sample decaying after 3 days?
Example 2: Exponential Growth
A bacterial culture has an initial population of 2000 bacteria. The predicted population of the culture
after 3 ℎ is 10 000 bacteria.
a) Assume that the bacteria population grows exponentially. Find an exponential function that models
the population growth.
b) Find the number of bacteria after 2 ℎ.
c) Find the rate of growth of the number of bacteria after 2 ℎ.
d) When will the bacteria population reach 18 000?
Example 3: Modelling of Movement
An automatic door has been programmed so that the angle, in degrees, that the door is open is
determined by the equation 𝑎(𝑡) = 180𝑡(2)–7 , where 𝑡 is the time in seconds, measured from when a
button is pressed.
a) Find an equation that models the rate at which the angle is changing with respect to time.
b) How quickly is the door closing after 5 𝑠?