rates of change and particle motion i if asked to … maximus page 1 of 10 rates of change and...

Calculus Maximus

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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

bojja_000

Stamp

Calculus Maximus WS 4.5: Rates of Change & Part Mot I

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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.

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

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

t

x

x'(t) = v(t)

right

negative

position

instantaneous

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

velocity

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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≤ ≤ .

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

& continuous

v(0) = -3 < 0 ,so the particle is moving in the negative direction, to the left.

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

Since v(t) is differentiable, it is also continuous.

Since v(0) < 0 and v(2) = 2 > 0 , by the IVT, v(t) = 0 at some time, thus the particle is at rest for some time t (0,2) [0, 12]

(c) Use data from the table to find an approximation for vʹ′(10) and explain the meaning of vʹ′(10) 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.

Since t = 0 is not in the table, we must use the two data points on either side of t = 0.

v'(10) = a(10) meters/min/min = -0.5 meter/min2

must show "squiggles

must show difference quotient


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(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.

v(4) = 6 > 0, so the particle is moving to the right at t = 4 seconds.

The particle is moving left on t ∈ (4, 9] seconds since v(t) < 0 on this interval

Since v(t) is positive and is decreasing at t = 4 seconds, v'(4) = a(4) < 0 , so the acceleration of the particle is negative at t = 4 seconds.

Avg accel = v(4) - v(2)4 - 2 = - 1.5 ft/sec2 show the difference quotient

Even though the average acceleration on t ∈[2,4] sec. is -1.5, the slopes of v(t) are undefined at t = 3 (sharp turn), so the graph of v'(t)=a(t) is not continuous for t ∈[2,4], so there is no garantee that v'(t)=-1.5 for some t ∈[2,4]

The particle is farthest to the right at t=5 seconds because v(t) >0 ∀ t ∈[0,5) and v(t) < 0 ∀ t ∈(5,9]. The particle therefore, changes direction from right to left at t = 5 seconds.

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 __________________.

3t - 12 t + 9 = 3(t - 3)(t -1) = 0

2

x'(0) = v(0) = 9 >0, so particle is moving to the right at t=0.

x''(t) = v'(t) = a(t) = 6t - 12 v'(1/2) = a(1/2) = -9 < 0So the velocity of the particle is decreasing at t = 1/2

x'(1/2) = v(1/2) = 3.75 > 0 a(1/2) = -9 < 0 Since v(1/2) > 0 and a(1/2) < 0 , the speed of the particle is decreasing at t=1/2.

particle moves left when v(t) < 0

x'(t) = v(t) = 3t - 12 t + 9

2

t = 3, 1

v(t) < 0 for t ∈ (1, 3)

Distance = |x(1) - x(0)| + |x(3) - x(1)| + |x(5) - x(3)| = 4 + 4 + 20 = 28 units

decreasing

decreasing

negative

negative

decreasing

positive


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.

3 2

speed is greatest at t = 8, since at this time, the graph of v(t) is farthest from the t axis.The speed at t = 8 is 6.

acceleration is greatest when the slope is most positive.v'(2) = a(2) = 0, v'(6 = a(6) < 0, v'(9) = a(9) > 0So acceleration is greatest at t = 9.

particle moves left when v(t) < 0, this occurs on t ∈ (4, 10)

speed is decreasing on t ∈ (2, 4) since v(t) > 0 and decreasing on this intervaland on t ∈ (8, 10), since v(t) < 0 and increasing on this interval.

Area ~ (y-axis units)(x-axis units) which would be vt or in units (ft/sec)(sec) = ft.we conclude that these areas give positive distances, in feet, and when added togather, their sum gives the total distance travelled by the particle in time interval [0, 11] = 19.5+ 2π


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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.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

2:00 pm t = 2

s'(t) = v(t) = 1 - 2t

s'(2) = v(2) = -3 so storm will be moving at 3 mph moving away from NB

_______________

_______

F

A

C


_____ (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

first time is t = 1.4792731 hours = 1 hr 28 min 45.3866 sec So, Mr. Wenzel reached 55 mph for the first time after he left at noon at about 1:28 pm.

B

at t = 8 sec, the stone is at 56 ft on the way upat t = 14 sec, the stone is at 56 ft on the way down.

h'(t) = v(t) = 11 - t

h'(8)= v(8) = 11 - 8 = 3 ft/sec

E

rates of change and particle motion i if asked to … maximus page 1 of 10 rates of change and...

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