5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:5

324 STEP 4. Review the Knowledge You Need to Score High

Leakage Problem

Example

Water is leaking from a faucet at the rate ofl(t)= 10 e−^0.^5 tgallons per hour, wheretis

measured in hours. How many gallons of water will have leaked from the faucet after a

24-hour period?

LetL(x) represent the number of gallons that have leaked afterxhours.

L(x)=

∫x

0

l(t)dt=

∫ 24

0

10 e−^0.^5 tdt

Using your calculator, enter

∫

( 10 e∧(− 0. 5 x),x,0,24)and obtain 19.9999. Thus, the

number of gallons of water that have leaked afterxhours is approximately 20 gallons.

TIP • You are permitted to use the following four built-in capabilities of your calculator to

obtain an answer: plotting the graph of a function, finding the zeros of a function,

finding the numerical derivative of a function, and evaluating a definite integral. All

other capabilities of your calculator can only be used tocheckyour answer. For example,

you maynotuse the built-in [Inflection] function of your calculator to find points of

inflection. You must use calculus using derivatives and showing change of concavity.

Growth Problem

Example

On a farm, the animal population is increasing at a rate which can be approximately repre-

sented byg(t)= 20 +50 ln(2+t), wheretis measured in years. How much will the animal

population increase to the nearest tens between the third and fifth years?

LetG(x) be the increase in animal population afterxyears.

G(x)=

∫x

0

g(t)dt

The population increase between the third and fifth year will

=G( 5 )−G( 3 )

=

∫ 5

0

(

20 +50 ln( 2 +t)dt

)

−

∫ 3

0

(

20 +50 ln(2+t)dt

)

=

∫ 5

3

[ 20 +50 ln(2+t)]dt.

Enter

∫

( 20 +50 ln( 2 +x),x,3,5)and obtain 218.709.

Thus, the animal population will increase by approximately 220 between the third and fifth

years.