Rates of Change and Particle Motion I...Page 1 of 6 §—Rates of Change and Particle Motion I If f...

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§—Rates of Change and Particle Motion I

If f x represents a quantity, then f x or df

dxrepresents the instantaneous rate of change of that

functions. ( )f x may describe a particle’s position or its velocity, but f x can represent ANY quantity,

such as the area of a circle, the outdoor temperature, the amount of rainfall, number of people infected with a disease, etc. This is the true power of the derivative: it gives us a way to talk about how fast ANYTHING is changing at a single moment in time.

Example 1:The temperature T, in degrees Fahrenheit, of a cold yam placed in a hot oven is given by )(tfT , where tis the time in minutes since the yam was put in the oven.a. What is the practical meaning of the equation 20 255f

b. What is the sign of )(tf ? Why?

c. What are the units of )20(f ?

d. What is the practical meaning of the statement 2)20( f ?

Example 2:An ice cream company knows that the cost, C (in dollars), to produce q quarts of cookie dough ice cream is a function of q, so )(qfC .a. If 70)200( f , what are the units of the 200? What are the units of the 70? Explain clearly what this

equation is telling you.

b. If 3)200( f , what are the units of the 200? What are the units of the 3? Explain clearly what thisequation is telling you.

Calculus Maximus Notes: 4.5T ROC & PM (\I

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Example 3:Let )(tf be the number of centimeters of rainfall that has fallen since midnight, where t is the time in hours. Interpret the following in practical terms, giving units.a. 1.3)10( f

b. 16)10(1 f

c. 4.0)8( f

d. 2)5(1 f

The word “instantaneous” is often used, even when x does not represent time. The word is actually often omitted. In practice, when we merely say “rate of change,” we mean “instantaneous rate of change.”

Example 4:The radius of a circle is increasing. How fast is the area of the circle changing with respect to the radius when the radius is a) 5 inches, and b) 10 inches? How fast is the circumference changing at these times?


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There is perhaps no more important type of rate of change than that of motion. After all, this was Newton’s motivation for inventing Calculus to begin with. We have already looked at motion briefly. Here is a summary with some new terminology.

s t v t a t , where s t is the position function.

Displacement over an interval ,a b : Displacement s s b s a

Distance traveled is the sum of the absolute values of the distances between resting points. Average velocity over an interval ,a b :

Average velocity displacement

slope of secant lineelapsed time

s b s a

b a

(instantaneous) velocity at time t : s t v t = slope of tangent line

Speed velocityds

v tdt

At time t c , if both 0v c and 0a c OR if both 0v c and 0a c , then speed is

increasing. At time t c , if 0v c and 0a c OR if 0v c and 0a c , then speed decreasing.

At time t c , if 0v c , position (distance) is increasing. If 0v c , position (distance) is

decreasing. At time t c , if 0a c , velocity is increasing. If 0a c , velocity is decreasing.

Example 5:A particle moves along a horizontal line so that its position at any time 0t is given by the function

2 4 3s t t t , where s is measured in feet and t is measured in seconds.

a. Find the displacement of the particle during the first 2 seconds.

b. Find the average velocity of the particle during the first 4 seconds.

c. Find the instantaneous velocity of the particle when 4t .

d. Find the acceleration of the particle when 4t .

e. At 4t , is the speed of the particle increasing or decreasing? Justify.

f. When is the particle moving to the right? Left?. At what values of t does the particle change direction?

g. Find the total distance the particle travels during the first 4 seconds.


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Example 6:The graph at right shows the position s t of a

particle moving along a horizontal line.a. When is the particle moving to the left?

b. When is the particle moving to the right?

c. When is the particle standing still?

d. During the first 6 seconds, what is the i)displacement? ii) distance traveled?

e. What is the particle’s velocity at 5t ?

f. Graph the particle’s velocity and speed (where it’s defined).

g. At what time is the particle moving the fastest?


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Example 7:The graph at the right shows the velocity v f t of

a particle moving along a vertical line.a. When is the particle moving down?

b. When is the particle moving up?

c. When does the particle reverse direction?

d. When is the particle moving at a constant speed?

e. When is the particle moving at its greatest speed?

f. When is speed increasing? Justify.

g. When is speed decreasing? Justify.

h. Graph the acceleration (where defined).

g. When is the acceleration the greatest?


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Example 8:The values of the coordinate s of a moving body along a line for various values of t are given below.

t (sec) 0 0.5 1 1.5 2 2.7 3 3.6 4s (ft) 40.0 35.0 30.2 36.0 48.2 45.0 38.2 16.0 0.2

Estimate the velocity at each of the following. Show the difference quotient!!

a) At 0.5t sec b) At 2.7t sec c) At 3.5t sec

Example 9:In Economics . . . Marginal cost is loosely defined as the extra cost of producing one more unit. Marginal revenue is the extra money brought in from selling one more unit. Marginal profit is the extra profit from making and selling one more unit. Profit Revenue Cost or P R C

Average Cost C xC x

x dollars per unit

Suppose it costs 3 26 15C x x x x dollars to produce x radiators when 8 to 10

radiators are produced, and that 3 23 12R x x x x gives the dollar revenue

from selling x radiators. Your shop currently produces 10 radiators a day.

a) What is the average cost of producing 10 radiators?

b) What is the profit of producing and selling 10 radiators?

c) What is the marginal cost to produce one more unit?

d) Show that the marginal cost when 10 radiators are produced is approximately the cost of producing onemore radiator after 10 have been made, by calculating the latter cost directly.

e) What is the marginal revenue of selling one more unit?

f) Should your company begin producing and selling 11 radiators daily instead of 10? Why or why not?What other factors might affect your decision?

Almost done!Hang in there.

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Name_________________________________________ Date________________________ Period______

Rates of Change and Particle Motion I

Show all work. No calculator unless otherwise stated. If asked to “Explain your answer,” write in

complete sentences.

For questions 1-5, answer in complete sentences. Be sure to start your sentence, when appropriate, with an introductory adverbial clause referencing the moment in time.

1. Let )(xf be the elevation in feet of the Mississippi river x miles from its source. What are the units of)(xf ʹ′ ? What can you say about the sign of )(xf ʹ′ ?

2. An economist is interested in how the price of a certain item affects its sales. Suppose that at a price of$p, a quantity, q, of the item is sold. If )( pfq = , explain the meaning of each of the followingstatements:

(a) 2000)150( =f

(b) 25)150( −=ʹ′f

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3. Suppose )(xg is the amount of time, in minutes, it takes to boil a cup of water if the water has an initialtemperature of x, degrees Fahrenheit.

(a) What are the units of )(xg ʹ′ ?

(b) What is the sign of )50(g ʹ′ ? Why?

4. If )(vg is the fuel efficiency, in miles per gallon, of a car going at v miles per hour, what are the unitsof )55(g ʹ′ ? What is the practical meaning of the statement 54.0)55( −=ʹ′g ?

5. Suppose that ( )f T is the cost of heating your house, in dollars per day, when the temperature outside isT degrees.

(a) Explain in practical terms the meaning of (23) 0.21f ʹ′ = − .

(b) If (23) 2.94f = and (23) 0.21f ʹ′ = − , approximate the cost to heat your house when the temperature is 20 degrees. Explain how you determined this cost.

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6. If ( )x t represents the position of a particle along the x-axis at any time, t, then the following statementsare true.

(a) “Initially” means when _________________________________ 0= .

(b) “At the origin” means _________________________________ 0= .

(c) “At rest” means _________________________________ 0= .

(d) If the velocity of the particle is positive, then the particle is moving to the ______________.

(e) If the velocity of the particle is _________________, then the particle is moving to the left.

(f) To find average velocity over a time interval, divide the change in _____________________ by the change in time.

(g) __________________________ velocity is the velocity at a single moment (instant) in time.

(h) If the acceleration of the particle is positive, then the ____________________ is increasing.

(i) If the acceleration of the particle is ____________________, then the velocity is decreasing.

(j) In order for a particle to change directions, the ____________________ must change signs.

(k) One way to determine _________________ _____________________

__________________ over a time interval, when given the position function or graph, is to find the sum of the absolute values of the differences in position between all resting points.

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7. If the position of a particle along a horizontal line is given by ( ) 3 21 3 43

x t t t t= − − + , find the total

distance traveled on the interval 0 6t≤ ≤ .

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8. The data in the table below gives selected values for the velocity, in meters/minute, of a particle movingalong the x-axis. The velocity v is a differentiable function of time t.

Time t (min) 0 2 5 6 8 12 Velocity ( )v t (meters/min)

3− 2 3 5 7 5

(a) At 0t = , is the particle moving to the right or to the left? Explain your answer.

(b) Is there a time during the time interval 0 12t≤ ≤ minutes when the particle is at rest? Explain your answer.

(c) Use data from the table to find an approximation for ( )10vʹ′ and explain the meaning of ( )10vʹ′ in terms of the motion of the particle. Show the computations that lead to your answer (this means show the difference quotient!!!!!), and indicate units of measure.

(d) (Genius Question) Let ( )a t denote the acceleration of the particle at time t. Is there guaranteed to

be a time t c= in the interval 0 12t≤ ≤ such that ( ) 0a c = ? Justify your answer.

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9. The graph below represents the velocity v, in feet per second, of a particle moving along the x-axis overthe time interval from 0t = to 9t = seconds.

(a) At 4t = seconds, is the particle moving to the right or left? Explain your answer.

(b) Over what time interval is the particle moving to the left? Explain your answer.

(c) At 4t = seconds, is the acceleration of the particle positive or negative? Explain your answer.

(d) What is the average acceleration of the particle over the interval [ ]2,4t∈ ? Show the computations that lead to your answer, and indicate units of measure.

(e) Genius Question) Is there guaranteed to be a time t in the interval [ ]2,4t∈ such that

( ) 1.5v tʹ′ = − 2ft/sec ? Justify your answer.

(f) At what time t in the given interval is the particle farthest to the right? Explain your answer.

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10. A particle moves along the x-axis so that at time t, its position is given by ( ) 3 26 9 11x t t t t= − + + .(a) At 0t = , is the particle moving to the right or to the left? Explain your answer.

(b) At 1/ 2t = , is the velocity of the particle increasing or decreasing? Explain your answer.

(c) At 1/ 2t = , is the speed of the particle increasing or decreasing? Explain your answer.

(d) Find all values of t for which the particle is moving to the left.

(e) Find the total distance traveled by the particle over the time interval 0 5t≤ ≤ .

11. Fill in the blanks.(a) If velocity is negative and acceleration is positive, then speed is _____________________.

(b) If velocity is positive and speed is decreasing, then acceleration is _____________________.

(c) If velocity is positive and decreasing, then speed is ________________________.

(d) If speed is increasing and acceleration is negative, then velocity is _______________________.

(e) If velocity is negative and increasing, then speed is ____________________________.

(f) If the particle is moving to the left and speed is decreasing, then acceleration is

________________________.

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12. The graph below shows the velocity, ( )v t , of a particle moving along the x-axis for 0 11t≤ ≤ . Itconsists of a semi circle and two line segments. Use the graph and your knowledge of motion toanswer the following questions.

(a) At what time, 0 11t≤ ≤ , is the speed of the particle the greatest?

(b) At which of the times, 2t = , 6t = , or 9t = , is the acceleration of the particle greatest? Explain your answer.

(c) Over what time intervals is the particle moving to the left? Explain your answer.

(d) Over what time intervals is the speed of the particle decreasing? Explain your answer.

(e) Find the area of the semicircle on the interval 0 4t≤ ≤ bounded by the curve and the x-axis, then find the area of the triangle on the interval 4 10t≤ ≤ bounded by the curve and the x-axis, and finally, find the area of the triangle on the interval 10 11t≤ ≤ bounded by the curve and the x-axis. If all of these areas were positive and added together, propose what quantity this might be in terms of the particle’s movement.

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Multiple Choice

_____ 13. A violent storm system is in the Central Texas area and moving in a straight line through New Braunfels. The Weather Service Office forecasts that t hours after mid-day, the storm will be a distance of 2( ) 12s t t t= + − miles from New Braunfels. How fast will the storm be moving at 2:00 pm, and in what direction will it be moving, respectively? (A) -5 mph, towards NB (B) 5 mph, towards NB (C) 5 mph, away from NB (D) -3 mph, towards NB (E) 3 mph, towards NB (F) 3 mph, away from NB

_____ 14. Find the rate at which the surface area of a sphere ( )24A rπ= is changing with respect to its

radius r when 4r = cm. (A) 232 cm /cmπ (B) 236 cm /cmπ (C) 227 cm /cm (D) 28 cm /cm (D) 248 cm /cm

15. It took Mr. Wenzel from noon until 7 pm to drive from NB to his in-laws’ house North of Dallas 385miles away. After t hours of driving, his distance from NB was given in miles by

2 3165 110( )7 49

s t t t= − .

_____ (i) What was his average speed for the trip to his in-laws’?

(A) 53 mph (B) 54 mph (C) 55 mph (D) 56 mph (E) 57 mph (F) 58 mph

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_____ (ii) (calculator OK) Mr. Wenzel’s instantaneous speed twice coincided with his average speed. At what time did it first happen?

(A) 1:33 pm (B) 1:28 pm (C) 5:41 pm (D) 5:31 pm (E) 1:38 pm

_____ 16. (Calculator Permitted) If a rock is thrown vertically upwards from the surface of the planet

Newton with an initial velocity of 11 ft/sec, its height after t seconds is given by 21112

h t t= − .

Find the velocity of the stone after it has risen 56 feet. (A) -1 ft/sec (B) 2 ft/sec (C) 0 ft/sec (D) 1ft/sec (E) 3 ft/sec

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