Differential Calculus Application Problems

8/13/2019 Differential Calculus Application Problems

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1 Understand the methods so you can solve similar problems.Understand the concepts so you can solve unfamiliar problems.

Study the (a) class notes, (b) text examples, (c) do the text exercises, and (d) do the 4 thhour problems

Differential Calculus

Applications


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Related Rates

As your morning coffee pours into your cup, the scintillating scent arouses your awareness. At

the same time both the volume of the coffee in the mug, and the height of coffee in the mug are

changing. Focus your awakening awareness on the rates of change of volume and height, andyou begin to sense the related rates ideas.

In this applicaton area, Rate of changerefers to the change of one variable (or measurement)

with respect to time. Related ratesconcerns the relationship between the time rates of two or

more variables.

height volume

slope is the

rate of change in the slope is the

height of coffee wrt time rate of change in the volume of coffee wrt time

To simplify, suppose your coffee mug is a cylinder of diameter 6 cm. and height 11 cm. The

radius,

r= 1/2 (diameter) = 1/2 (6) = 3 cm.and when the height of coffee in the mug is hcm., the volume of coffee in the mug is

V = r h h h2 23 9= =

so with respect to time t,

dV

dt

dh

dt= 9

relates the rate of change in volume to the rate of change h

in height.

time time


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EXAMPLE 1

A gas is pumped into a spherical balloon at a constant rate of 25 cubic cm. per second. The

balloon does not pop during this question, and the increase in the volume of the balloon equals

the volume of gas put in the balloon. How fast is the radius of the balloon increasing when the

radius is 8 cm.?

Given:dV

dt= 25 cm3 /sec Diagram and variable identification:

r

Want:dr

dt r cm=8 .

V = volume, r = radius

Relationship:The relationship relates the variables under Givenand Want.

V =4

3

3r

Differentiate: Always differentiate with respect totime tin related rates questions.

dV

dtr

dr

dt

rdr

dt

=

=

4

33

4

2

2

Evaluate: When r = 8 anddV

dtcm= 25 3 / sec

25 = 4 8

25

256 0 031

2

8

dr

dt

dr

dtr cm==

.

Answer:The radius of the balloon is increasing at 0.031 cm/sec when the radius is 8 cm.

Caution: Never use the condition under "Want" until

you have differentiated. We use r = 8 in theEvaluate

step after we have differentiated.


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EXAMPLE 2

Wheat from prairie farms has arrived by train at the grain elevator, and is being unloaded. The

wheat is falling at 5 cubic metres per minute onto a pile in the shape of an inverted cone. The

cone always has a ratio of diameter to height of 3 to 2. How fast is the height of the cone

increasing when the diameter is 8 m.?

Diagram and variable identification:

Given:

Want:

Relationship: Relate the variables in Givenand Want.

(Relate volume to height; eliminate radius from the relationship.)

Differentiate: Always differentiate with respect to time t in related rates questions.

Evaluate:

Answer:


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EXAMPLE 3

With the approximately flat prairies, cars can maintain constant speeds relatively easily. Our

story commences one morning when Bill's car heads north from Calgary at 10 am at a constant

speed of 80 km per hour. At noon Mary heads west from Medicine Hat at a constant speed of 95

km per hour. How fast is the distance between Bill and Mary changing at 1:30 pm? Medicine

Hat is 200 km east of Calgary.


Given:

Want:


Differentiate: Always differentiate with respect to time tin related rates questions.

Evaluate:

Answer:

N

Medicine HatCalgary


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EXAMPLE 4

A ferris wheel of radius 10 m. is rotating counter clockwise a full circle every 12 minutes. The

center of the ferris wheel is 10.3 m. above ground. Natasha sits on one chair on the ferris wheel.

Find the rate in metres per hour at which the height of Natasha's chair is changing when the chair

is 14 m. above the ground.


Given:d

dthr hr

= =5 2 10/ /

Want:dy

dt y m=14 .


( )y= +10 3 10. sin


( )dy

dt

d

dt= 10cos

Evaluate: When ( )y= + = = 14 10 3 10 14 0 37 0 379, . sin( ) , sin . , .

dy

dt y m==

14

10 0 379 10

2919

cos( . )

.

Answer: The height of Natasha's chair is increasing at 291.9 m/hr when it is 14m above the ground.

10sin( )

.3

10.3

y

is the angle in radiansmeasured from the horizontal

y is the height of Natasha's chair


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EXAMPLE 5

Consider the wheel-piston diagram below. A wheel of radius 1 cm. revolves counterclockwise at

500 rotations per second. The connecting rod to the piston is 5 cm. long. How fast is the height

of the piston changing when the angle of elevation is

=6

? Give your answer in metres per

second.


Given:

Want:

Relationship:


Evaluate:

Answer :

5

piston

1


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EXAMPLE 6

A swimming pool is 9 m. at the deep end, 2 m. at the shallow end, 10 m. long and 5 m. wide.

Water is entering the pool at the rate of 8 cubic metres per minute. How fast is the height of

water in the pool rising when the height of water in the pool is (a) 3m. (b) 8 m. ?


Given:

Want:

Relationship: (Caution, your model here is a piecewise function with different pieces for parts (a) and (b) ).

Differentiate:

Evaluate:

Answer:(a) (b)


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EXAMPLE 7

A fisher is reeling in a fish. She is raising the tip of her fishing rod at the rate of 30per second,

and reeling the line in at 10 cm. per second. The fishing rod is 2 metres long. How fast is the

fish approaching shore when the tip of the fishing rod is 1.1 metres above the water and 20

metres of line go from the tip of the rod, to the fish at the surface of the water?


Given:

fishing line

fishing rod

Want:

fish

Relationship:

Differentiate:

Evaluate:

Answer:


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Fourth 4.1, Related Rates

A water trough with cross sections of inverted isosceles triangles is 4 m long, 1.5 m wide at the

top and 1 m deep. If water is being pumped out at the rate of 0.1 m

3

per minute, how fast is thedepth of the water changing when the depth is 0.6 m?

top


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Absolute Maximum and Minimum Values

1. Letf(x) = x1/3(x+3)2/3. (a) Analytically find all the critical numbers of f.

(b) Analytically find the absolute maximum value and the absolute minimum value off on the

interval [-2,2]. Support your answer.

2. Analytically find the exact absolute maximum and absolute minimum values of

g(x) = x / (x2+ 9) on [-2,6].


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3. The function f(x) = 2 sin2(x) + 2 3 cos(x) 2 sin(x) + x 3 - 4

has derivative )3)cos(2)(1)sin(2()( = xxxf .

Find the critical numbers of the function.

Find the absolute maximum and minimum values of f on the interval [ -/4, /2 ].

4. Find the exact absolute maximum and minimum values of f(x) = ln(x)/x on [0.1 ,6].


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Fourth 4.2

1. Use a graph to estimate the absolute extrema ofy = cos(sin(x))on [0,] and then use analytic

methods to find the exact values.

2. Find and classify the critical numbers of f given3 4

35)(

x

xxf

= .

3. Suppose f(x) = xa(3-2x)

b and 0 < b < aand 0 x 3/2. Find the maximum value of f.


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Derivatives and the Shape of Curves

1. (a) Analytically determine the exact critical numbers of 2)( 3 53 = xxxg .

(b) Use an appropriate derivative test to classify any local extrema. Clearly state your

conclusions.

2. Find a cubic function dcxbxx

xf +++= 23

3)( that has a local maximum atx= - 2, a

local minimum atx= 1, and f(0) = 1.


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3. Circle the function(s) for which the Mean Value Theorem applies and cross out the

function(s) for which the Mean Value Theorem does not apply:

(a) f(x) = x2/3

on [-2,2] (b)2

1)(

+=

x

xxf on [-2,2] (c)

3

2)(

+

+=

x

xxf on [-2,2].

4. Determine the exact value(s) ofxwheref(x) = x2e

-x is increasing most rapidly. Begin with a

sketch of

y = f(x).


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Stories told by the 1stand 2

ndderivatives

1. Find the intervals of increase and decrease and the local extrema of

f(x) = x4 4x

3- 12x

2+ 32x + 5

2. Find the intervals of concavity and inflection points of y = x4 6x

2+ 1.

3. Use the Second Derivative Test to find the local extrema off(x) = x5 3x3+ 4.

4. The graph of the derivative f is sketched to the right. Use it to answer questions about the

functionf. Use estimates where an exact answer is not available.

Forf, find the intervals of increase:

Where isfconcave up?

For what values ofxdoesf have an inflection point?

a vertical tangent?-4 -3 -2 -1 1 2 3 4

x

y


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5. Estimate the inflection points and intervals of concavity of y = x2/3

cos(x)on [-/2,2].

[Hint: calculator question.]

6. Sketch the graph of a function satisfying all the following conditions:

xwhenxfxwhenxfxfxfxfxxx


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Fourth 4.3

1. For the function2)3(

)(

=x

xxf find (a) the intervals on whichf is increasing;

(b) the intervals on whichf is concave upwards;

(c) the inflection points off;

(d) the maximum value off on the interval [-8,2].

2. For the function f(x) = cos2(x) + 2 sin(x)find the following

(a) The critical numbers;

(b) The intervals of increase;

(c) The inflection points.


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LHopitals Rule

++

125

3lim

2

2

xx

x

x

))ln(lim 20

xxx

+

x

x

x

)sin(lim

0

x

x

x

)cos(1lim

0

x

x x

65

1lim

+

x

x x

4

2

31lim

+

( ))cot()ln(lim1

xxx


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x

xx /1)3(lim +

2lim

x

ex

x

+ )ln(

1

1lim

1 xx

x

x

)tan(

)sin(lim

0 xx

xx

x

x

xx

x 12

38lim

0


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Fourth 4.5

1. Find the limits: (a)

5

)ln(lim

x

x

x

(b))cos(1

0lim x

xx

+

(c) ))(tan(lim 12

xe xx

2. Let S(r)be the surface area of an open cylinder (which has both a top and a bottom) of height

3.6 m and radius r, and let V(r)be the volume of the cylinder. Find

+ )(

)(lim

0 rV

rS

r.


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Optimization1. Find the volume of the largest right circular cone that can be inscribed in a sphere of

radius 5 m.

2. A trough of length 30 m is formed by scoring a rectangle of sheet metal 30 x 6 m and folding

up sides of 2 m width. Find the angle which gives a maximum volume for the trough.

3. A 2 m wall stands 6 m from a building. Find the length of the shortest ladder than will reach

the building from the ground outside the wall.


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Fourth 4.61. Find the dimensions of the rectangle of largest area that has its base on thex-axis and its other

two vertices above thex-axis on the curvey = 24 x2.

2.x

xx /2

0)31(lim

3. Find to 3 decimal accuracy the inflection points of f(x) = cos(2x) sin(5x)on the interval

(0,/3).

4. Coffee is flowing through a conical filter at the rate of 30 cm3per minute. The filter is 20 cm

high and has a 20 cm diameter at the top. How fast is the level of coffee in the filter falling whenthe level is 8 cm?

Differential Calculus Application Problems

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