Exponential Exercises

The value of a car is V(x)= 10(.75)^x, 
where x>=0 is the age of the car in years and 
V(x) is its value in thousands of $.
Rewrite the function using a rational number instead of the decimal. 
Is it an increasing or decreasing function?
What is the starting value of the car?

Evaluate:
V(0)
V(1)
V(2)
V(3)
V(5)
V(10)
V(20)

Solve:
V(x)=10
V(x)=5
V(x)=2.5
V(x)=1.25
Based on these, how much time does it take for the value of the car to halve?

Graph the function. What is the y-intercept?

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The price of a stock of MEGA Corp. is P(x)= 10(1.25)^x, 
where x>=0 is the numbers of years from today and 
P(x) is the stock's price in $.
Is it an increasing or decreasing function?
What is the starting value of the stock?

Evaluate:
P(0)
P(1)
P(2)
P(3)
P(5)
P(10)
P(20)

Solve:
P(x)=10
P(x)=20
P(x)=40
P(x)=80
Based on these, how much time does it take for the price of the stock to double?

Graph the function. What is the y-intercept?

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Some 60°C warm stuff is put into a 10°C cool place. Its temperature T after t minutes
is T(t)= 10 + 50e^(-0.1t)
Is it an increasing or decreasing function?
Evaluate:
T(0)
T(1)
T(2)
T(5)
T(10)
T(20)
T(30)
T(50)

What looks to be the horizontal asymptote as t gets large?

Solve:
T(t)=60
T(t)=50
T(t)=40
T(t)=30
T(t)=20
T(t)=15
T(t)=11
T(t)=10.5
T(t)=10.1
T(t)=10.01
T(t)=10

Graph the function. What is the y-intercept?

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World population doubled in the 40 years between 1960 and 2000, 
from 3 billion to 6 billion.
From this fact, a function that predicts future world population, in billions, is:
  WP(t)= 6*2^(0.025t)
where t is years since 2000. 
Is it an increasing or decreasing function?
Evaluate:
If this trend continues (it hasn't but pretend it did),
what should the world population be 
in 2021?
in 2000? 
in 2050?
in 2100?

Solve:
When will the world population be 
 10 billion? WP(t)=10
  6 billion? WP(t)=6
100 billion? WP(t)=100

Graph the function. What is the y-intercept?

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Carbon-14 is radioactive, which means some random atoms of a clump
of it will change to some other kind of atom at a known rate. 
Half of it will have changed in 5730 years (its "half-life").
The decimal percent amount of the starting pile that still is carbon-14
after t years is:
  S(t)= e^(-0.000121t)
Is it an increasing or decreasing function?                       
Evaluate:
How much is still carbon-14 after 
 0 years: S(0)
 1000 years: S(1000)
 2000 years: S(2000)
 10000 years: S(10000)
 20000 years: S(20000)
 100000 years: S(100000)

Solve:
How many years for there to be one half of the starting amount: S(t)=.5
How many years for there to be one quarter of the starting amount: S(t)=.25
How many years for there to be one eighth of the starting amount: S(t)=.125
How many years for there to be one sixteenth of the starting amount: S(t)=.0625
Based on these, how much time does it take for the amount to halve?
Graph the function. Adjust the X-axis.