Applications of Derivatives PDF Differentiation implicitly with respect to

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APPLICATIONS OF DERIVATIVES

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ANSWERS TO ODD-NUMBERED EXERCISES. 17. 3.4. Answers to Odd-Numbered Exercises ... derivatives of all orders. 28.1.7. Convention. When we say that a function f ...

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CHAPTER 3 APPLICATIONS OF DERIVATIVES

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In this chapter we will study applications of the derivative in various of a function that are necessary for the solution of many applied problems.

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APPLICATIONS OF DERIVATIVES

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2.7 Applications of Derivatives to Business and Economics

Of course we must often interpret answers to problems in light of the fact that x is

Applications of Derivatives

Differentiation implicitly with respect to t (or whatever the independent variable is). Note: It's okay to get a negative answer in these problems.

APPLICATION OF DERIVATIVES

Therefore local maximum value (–2) is less than local minimum value 2. Long Answer Type (L.A.). Example 11 Water is dripping out at a steady rate of 1 cu cm/

Applications of the Derivative

Such a problem differs in two ways from the local maximum and minimum problems we encountered when graphing functions: We are interested only in the function

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Functions Derivatives and their Applications Annuity

Applications of Derivatives

Related Rates

General steps

1. Draw a picture!! (This may not be possible for every problem, but there"s usually something you can

draw.)

2. Label everything. If a quantity is fixed for the entire problem, write in the number. If it can change,

then assign it a variable. There are often multiple ways to draw and label things, but the final answer

will be the same irrespective of how you label things.

3. Write down what you know, and what you want to know. Note: When writing down given/known rates

of change, make them positive if the variable is getting larger, negative if the variable is getting smaller

(this is going to depend on how you labeled your picture).

4. Figure out how everything is related and come up with a formula relating the variables. This can involve

using geometric formulas, triangles, similar triangles, etc.

5. Differentiation implicitly with respect tot(or whatever the independent variable is). Remember, in

these problems all variables are viewed as functions oft, so you"ll pick upd dtterms from the chain rule as you differentiate.

6. Plug in known values for variables and rates, then solve for the quantity in which you"re interested.

Warning: DO NOT plug in numbers for quantities that can change untilafteryou differentiate!! If a quantity is constant throughout the entire problem and cannot change, then you should have already put in the picture as a number in step 2. (Replacing a variablewith an expression involving another variableisallowed.)

Note:It"s okay to get a negative answer in these problems. A negative answer tells us that that the quantity

is decreasing and positive tells us it"s increasing. On a test or homework I want to see the correct sign in your

answer. In some books (not ours) all rates are positive in thefinal answers, even if the quantity is decreasing

(the negative may be assumed from inclusion of the word "decreasing" in the problem). 1

Problems

1. A spotlight on the ground shines on a building 36 ft away. A man 6 ft tall walks from the spotlight

toward the building at a speed of 10 ft/s. How fast is the length of his shadow on the building changing

when he is 24 ft from the building?

2. Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing

when the area of the square is 16 cm 2?

3. A point is moving along the graph ofy=2

3 +xsuch thatdx/dtis 4 centimeters per minute. Finddy/dt

for each value ofx.

4. A rectangular swimming pool is being filled with water at a rate of 5 m3/min. The length of the pool

is 10 m and the width is 4 m. How fast is the height of the water increasing?

5. Car A is 100 mi. east of Car B at 3:00pm. Car A is moving west at60 mi/h and Car B is moving south

at 70 mi/h. How fast is the distance between the cars changingat 4:00pm?

6. A balloon rises at a rate of 3 m/s from a point on the ground 30m from an observer. Find the rate of

change of the distance between the observer and the balloon when the balloon is 30 m above the ground.

7. A company that manufactures sport supplements calculates that its costs and revenue can be modeled

by the equations

C= 125,000 + 0.75xandR= 250x-1

10x2

wherexis the number of units of sport supplements produced in 1 week. If production in one particular

week is 1000 units and is increasing at a rate of 150 units per week find: (a) The rate at which the cost is changing. (b) The rate at which the revenue is changing. (c) The rate at which the profit is changing. 2

Answers

1.-2.1ft/s

2. 48cm2/s

3. dy dt=-8(3+x)2 4. + 1

8m/min

5. +31 mi/h

6. The picture we drew in class formed a right triangle with base 30, heighty, and hypotenusex, so the

equation relating them isx2= 302+y2. We are given thatdy dt= +3 , and we want to know whatdxdt is wheny= 30 (note: wheny= 30,x=⎷

302+ 302=⎷1800≈42.43). Differentiating the equation

yields 2xdx dt= 0 + 2ydydt Plugging in all the values we know and solving for dx dtgives us:

2·42.43·dx

dt= 2·30·3 dx dt=9042.43≈2.12 m/s

7. For all parts, we"re finding the rates for the particular week referenced in the problem, whenx= 1000.

There aren"t any pictures to draw for this problem, and the equations are already given to us. We"re also given that dx dt= +150. So, all we need to do is differentiateC,R, and

P=R-C= 250x-1

10x2-(125000 + 0.75x) =-110x2+ 249.25x-125000

and plug in our values forxanddx dt: (a) dR dt= 250·dxdt-110·2·x·dxdt-→dRdt= 250·150-15·1000·150 =7500 (b) dC dt= 0.75·dxdt-→dCdt= 0.75·150 =112.5 (c) dP

dt=-110·2·x·dxdt+ 249.25·dxdt-→dPdt=-110·2·1000·150 + 249.25·150 =7387.5

3 Increasing and Decreasing FunctionsDefinition 1.Letf(x) be a function defined on an intervalI. Informally, this means that we"re moving uphill (as we travel in the positivexdirection). •f(x) is said to bedecreasingonIiff(x1)≥f(x2) for any two numbersx1,x2inIsuch thatx1< x2. Informally, this means that we"re moving downhill (as we travel in the positivexdirection). (a)f(x) is increasing(b)f(x) is increasing(c)f(x) is decreasing(d)f(x) is decreasing Figure 1: Some Examples of Increasing and Decreasing Functions

Definition 2(Critical Numbers).Acritical number(orcritical value)cof a functionf(x) is any number [in

the domain off] such thatf?(c) = 0 orf?(c) is undefined. Theorem 3("Test for Increasing and Decreasing Functions").Letf(x)be a differentiable function on the intervalI,

1. Iff?(x)>0for allxinI, thenf(x)isincreasingonI.

2. Iff?(x)<0for allxinI, thenf(x)isdecreasingonI.

3. Iff?(x) = 0for allxinI, thenf(x)isconstantonI.

How we use this

Finding intervals on which a function is increasing, and on whichit"s decreasing.

1. Find all critical numbers (x-values wheref?is zero or undefined)...

2. ...and plot them on a number line. This divides the number line into intervals.

3. Pick a test value from each interval (I"ll denote these with a star) and plug intof?to see if the derivative

is (+) or (-). Note: we don"t need the actual value off?at each test value - only the sign.

4. Interpret:

•fis increasing on the (+) intervals. •fis decreasing on the (-) intervals. 4

ExtremaDefinition 4.Letf(x) be defined atx=c.

1.f(c) is said to be arelative maximum(orlocal maximum) offif and only iff(c) is greater than the

surrounding function values, i.e.,f(c) is the largest function value "close to"x=c. (Looks like the peak of a mountain.)

2.f(c) is said to be arelative minimum(orlocal minimum) offif and only iff(c) is less than the

surrounding function values, i.e.,f(c) is the smallest function value "close to"x=c. (Looks like the bottom of a valley.) Definition 5.Letf(x) be defined on an intervalIcontainingc. •f(c) is said to be theabsolute maximumoffonIif and only iff(c)≥f(x) for allxinI.

How we use this

Finding the relative maxima and minima of a function. Theorem 6(The First Derivative Test).Supposecis a critical number of a [continuous] functionf,

1. Iff?changes from(+)→(-)(i.e., it goes from increasing to decreasing) atc, thenfhas a relative

maximum at(c,f(c))

2. Iff?changes from(-)→(+)(i.e., it goes from decreasing to increasing) atc, thenfhas a relative

minimum at(c,f(c))

3. Iff?does not change sign (i.e., we have(+)→(+)or(-)→(-)) or iff(c)is undefined, thenfhas

no relative maximum or minimum atc. Finding the absolute maximum and absolute minimum of a continuousfunctiony=f(x)on a closed interval[a,b].

0. Check to make sure the interval is closed (easy) and that the function is continuous on the interval (it"s

okay to have discontinuities outside the interval).

1. Find all critical values offin the interval(we don"t care about those that aren"t in the interval).

2. Evaluatef(notf?) at those critical points and ataandb(the endpoints of the interval).

3. The largest value from step 2 is the absolute maximum offon [a,b], the smallest value is the absolute

minimum offon [a,b].

Notes: When finding absolute maximum/minimum, we"re interested in the function values (y"s); for many

other problems in this chapter, we"re more interested in thex-values. The absolute maximum or minimum

may occur at more than onex-value. 5

ConcavityDefinition 7(Formal definition of concavity).Letfbe a differentiable function on some open intervalI.

1.fis said to beconcave uponIiff?is an increasing function onI.

2.fis said to beconcave downonIiff?is a decreasing function onI.

Definition 8(Informal definition of concavity)."Concave up, like a cup. Concave down, like a frown." Definition 9.A point wherefchanges from concave up to concave down or from concave down to concave up is called aninflection pointsor apoint of inflection(orinflection point). (a) A concave up function(b) A concave down function

Figure 2: Concavity Examples

(a) Inc. and Concave Up(b) Inc. and Concave Down(c) Dec. and Concave Up(d) Dec. and Concave Down Theorem 10("Concavity Test" - this is what some people mean by "The Second Derivative Test"). •Iff??(x)>0for allxin an intervalI, thenf(x)isconcave uponI. •Iff??(x)<0for allxin an intervalI, thenf(x)isconcave downonI. Theorem 11(The Second Derivative Test).Supposef??(x)is continuous nearc,

1. Iff?(c) = 0andf??(c)>0, thenfhas a local minimum atc.

2. Iff?(c) = 0andf??(c)<0, thenfhas a local maximum atc.

3. Iff??(c) = 0, or iff?(c)orf??(c)is undefined, then this test is inconclusive.

How we use thisFinding local maxima and minima.The Second Derivative Test gives us another way of checking to see if a critical value corresponds to a local

maximum or local minimum.

Finding the Point of Diminishing Returns.

Iffchanges from concave up to concave down at an inflection point, then (in some contexts) that inflection

point is called the "point of diminishing returns".

Graphical Example

?6?4?22468 ?2 ?1 1 2 3 4 5 (Most of the following numbers are approximate and have beenrounded to 2 decimal places.)

Thecoordinatesof the

relative maximaare: (-5.52,4.97), (-2.29,1.90), (1.58,0.61), (5.97,1.68).

Thecoordinatesof the

relative minimaare: (-4,0), (0,0), (3.92,-1.55), (7,0).

Thecoordinatesof the

inflection pointsare: (-5.01,3.07), (-3.29,0.84), (-1.27,1.02), (0.8,0.3), (2.9,-0.56), (5.09,0.24), (6.61,0.69).

This function is increasing on theintervals(-∞,-5.52), (-4,-2.29), (0,1.58), (3.92,5.97), (6.61,∞).

This function is decreasing on theintervals(-5.52,-4), (-2.29,0), (1.58,3.92), (5.97,6.61). This function is concave up on theintervals(-5.01,-3.29), (-1.27,0.8), (2.9,5.09), (6.61,∞). This function is concave down on theintervals(-∞,-5.01), (-3.29,-1.27), (0.8,2.9), (5.09,6.61). 7

Optimization ProblemsGeneral steps:

1. Draw a picture and assign variables.

2. Write down the equation to be maximized or minimized (thisis sometimes called theobjective equation)

and the equation that describes the constraint (this is sometimes called theconstraint equation).

3. Use the constraint equation to rewrite the objective equation so that it has only one independent variable.

4. Find the domain of the [new] objective equation.

5. Find the maximum or minimum using calculus. Technically we"re finding an absolute max/min, but in

these problems it very often occurs at a local max/min in the domain (and in many problems there"s only one possibility).

6. Verify your answer using the Second Derivative Test.

Exercises

1. A farmer wants to build a pen with two dividers in order to separate elephants, donkeys, and penguins.

If 600 ft of fence is available and one side of the pen is bounded by a river and needs no fence since all

the animals just happen to have an irrational fear of water, then what is the maximum area that can be enclosed?

2. A carpenter wants to build a rectangular box with square sides in which to put round things. The

material for the bottom costs $20/ft

2, material for the sides costs $10/ft2, and the material for the top

costs $50/ft

2t. If the volume of the box must be 5 ft3, then find the dimensions that will minimize the

cost (and find the minimum cost).

3. A knight sees a damsel in distress 3 miles downstream on theopposite side of a straight raging river 0.5

miles wide. The knight can swim at 4 mi/hr and run at 7 mi/hr. Atwhat point on the opposite side should the knight swim in order to reach the distressed damsel as soon as possible.

4. After being rescued the distressed damsel decided to buy apeach orchard, and she wants to maximize

the number of peaches produced by her orchard. She has found that the per-tree yield is equal to 900

whenever she plants 45 or fewer trees per acre, and that when more than 45 trees are planted per acre,

the per-tree yield decreases by 25 peaches per tree for everyextra tree planted. For example, if there

were 40 trees planted per acre, each tree would produce 900 peaches. If there were 50 trees planted per

acre, each tree would produce 900-25(50-45) = 775 peaches. Find the number of trees that should be planted per acre to maximize the yield, and find the maximumyield per acre.

5. A 5 in×8 in piece of paper has a square cut out of each corner (same size from each) and is then folded

to make an open-top box. Find the size of the square that will maximize the volume.

6. Find the area of the largest rectangle that can be inscribed inside an isosceles triangle with side lengths⎷

2,⎷2,2.

7. A box with a square base and open top must have a volume of 32000 cm3. Find the dimensions of the

box that will minimize the amount of material needed.

8. Suppose Cheerwine

R?Bottling Company has asked you to design a cylindrical can that will hold 355 ml and uses the least amount of aluminium. What should be the dimensions of the can? 8quotesdbs_dbs21.pdfusesText_27

[PDF] Applications of Derivatives Differentiation implicitly with respect to

Applications of Derivatives

Related Rates

General steps

1. Draw a picture!! (This may not be possible for every problem, but there"s usually something you can

2. Label everything. If a quantity is fixed for the entire problem, write in the number. If it can change,

3. Write down what you know, and what you want to know. Note: When writing down given/known rates

4. Figure out how everything is related and come up with a formula relating the variables. This can involve

5. Differentiation implicitly with respect tot(or whatever the independent variable is). Remember, in

6. Plug in known values for variables and rates, then solve for the quantity in which you"re interested.

Problems

1. A spotlight on the ground shines on a building 36 ft away. A man 6 ft tall walks from the spotlight

2. Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing

3. A point is moving along the graph ofy=2

3 +xsuch thatdx/dtis 4 centimeters per minute. Finddy/dt

4. A rectangular swimming pool is being filled with water at a rate of 5 m3/min. The length of the pool

5. Car A is 100 mi. east of Car B at 3:00pm. Car A is moving west at60 mi/h and Car B is moving south

6. A balloon rises at a rate of 3 m/s from a point on the ground 30m from an observer. Find the rate of

7. A company that manufactures sport supplements calculates that its costs and revenue can be modeled

C= 125,000 + 0.75xandR= 250x-1

Answers

1.-2.1ft/s

2. 48cm2/s

8m/min

5. +31 mi/h

6. The picture we drew in class formed a right triangle with base 30, heighty, and hypotenusex, so the

302+ 302=⎷1800≈42.43). Differentiating the equation

2·42.43·dx

7. For all parts, we"re finding the rates for the particular week referenced in the problem, whenx= 1000.

P=R-C= 250x-1

10x2-(125000 + 0.75x) =-110x2+ 249.25x-125000

1. Iff?(x)>0for allxinI, thenf(x)isincreasingonI.

2. Iff?(x)<0for allxinI, thenf(x)isdecreasingonI.

3. Iff?(x) = 0for allxinI, thenf(x)isconstantonI.

How we use this

1. Find all critical numbers (x-values wheref?is zero or undefined)...

2. ...and plot them on a number line. This divides the number line into intervals.

3. Pick a test value from each interval (I"ll denote these with a star) and plug intof?to see if the derivative

4. Interpret:

ExtremaDefinition 4.Letf(x) be defined atx=c.

1.f(c) is said to be arelative maximum(orlocal maximum) offif and only iff(c) is greater than the

2.f(c) is said to be arelative minimum(orlocal minimum) offif and only iff(c) is less than the

How we use this

1. Iff?changes from(+)→(-)(i.e., it goes from increasing to decreasing) atc, thenfhas a relative

2. Iff?changes from(-)→(+)(i.e., it goes from decreasing to increasing) atc, thenfhas a relative

3. Iff?does not change sign (i.e., we have(+)→(+)or(-)→(-)) or iff(c)is undefined, thenfhas

0. Check to make sure the interval is closed (easy) and that the function is continuous on the interval (it"s

1. Find all critical values offin the interval(we don"t care about those that aren"t in the interval).

2. Evaluatef(notf?) at those critical points and ataandb(the endpoints of the interval).

3. The largest value from step 2 is the absolute maximum offon [a,b], the smallest value is the absolute

1.fis said to beconcave uponIiff?is an increasing function onI.

2.fis said to beconcave downonIiff?is a decreasing function onI.

Figure 2: Concavity Examples

1. Iff?(c) = 0andf??(c)>0, thenfhas a local minimum atc.

2. Iff?(c) = 0andf??(c)<0, thenfhas a local maximum atc.

3. Iff??(c) = 0, or iff?(c)orf??(c)is undefined, then this test is inconclusive.

Finding the Point of Diminishing Returns.

Graphical Example

Thecoordinatesof the

Thecoordinatesof the

Thecoordinatesof the

Optimization ProblemsGeneral steps:

1. Draw a picture and assign variables.

2. Write down the equation to be maximized or minimized (thisis sometimes called theobjective equation)

3. Use the constraint equation to rewrite the objective equation so that it has only one independent variable.

4. Find the domain of the [new] objective equation.

5. Find the maximum or minimum using calculus. Technically we"re finding an absolute max/min, but in

6. Verify your answer using the Second Derivative Test.

Exercises

1. A farmer wants to build a pen with two dividers in order to separate elephants, donkeys, and penguins.

2. A carpenter wants to build a rectangular box with square sides in which to put round things. The

2, material for the sides costs $10/ft2, and the material for the top

2t. If the volume of the box must be 5 ft3, then find the dimensions that will minimize the

3. A knight sees a damsel in distress 3 miles downstream on theopposite side of a straight raging river 0.5

4. After being rescued the distressed damsel decided to buy apeach orchard, and she wants to maximize

5. A 5 in×8 in piece of paper has a square cut out of each corner (same size from each) and is then folded

6. Find the area of the largest rectangle that can be inscribed inside an isosceles triangle with side lengths⎷

2,⎷2,2.

7. A box with a square base and open top must have a volume of 32000 cm3. Find the dimensions of the

8. Suppose Cheerwine