Newton’s Law of cooling
Standards: AII.F-IF.C.7eGraph functions expressed symbolically and show key features of the graph; by hand in simple cases and using technology for more complicated cases. Graph functions expressed symbolically and show key features of the graph; by hand in simple cases and using technology for more complicated cases. AII.F-LE. A4: Construct and compare linear, quadratic, and exponential models and solve problems. For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. PC.F-BF.B5:Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Part1: Knowing the basic properties of exponential and logarithmic functions.
Complete the following tables
f(x)=2^x
Base
Decay/Growth
Domain
Range
X-intercept
y-intercept
Vertical asymptote
Horizontal Asymptote
Inverse function
Sketch
f(x)=(1/3)^x
Base
Decay/Growth
Domain
Range
X-intercept
y-intercept
Vertical asymptote
Horizontal Asymptote
Inverse function
Sketch
f(x)=logx
Base
Domain
Range
X-intercept
y-intercept
Vertical asymptote
Horizontal Asymptote
Inverse function
f(x)=Ln(x-1)
Base
Domain
Range
X-intercept
y-intercept
Vertical asymptote
Horizontal Asymptote
Inverse function
Sketch
Part2: Converting exponential form into logarithmic form and vice versa:
(1/2)^(-4)=16 b) 〖log〗_27〖9=〗 2/3
Logarithmic form:………………… Exponential form: …………………………………..
Part3: Solving exponential and logarithmic equations.
3^(x-2)=〖27〗^x b) ⅇ^(x-2)=4
c) logx=-2 d) log_x100=2
Part 4: Mental math
Without using calculator, find
a) 〖log〗_8 64 b) 〖log〗_16 64 c) 〖log〗_(1/2) 8
Part5: Properties of logarithms
i) Condense 〖3log〗_2 4-〖log〗_2 8-〖log〗_2 2 into single logarithm and then evaluate.
(Show your work)
ii) Given log 2 =0.3010 and log3=0.4771. Use the given to find
(Answers only are not accepted, you need to show your work)
a) log 6=
b) log 8 =
c) log 30 =
Part5: Graphing exponential functions using intercepts and transformations.
Use intercepts and transformations to sketch the graph of y=-2(3)^(-(x+1) )+1
(Show your work)
x-intercept:
y-intercept:
Sequence of transformation:
Part6: Connection to physics: Newton’s Law of cooling:
For this part you need to use Newton’s law of cooling formula shown below.
With the aid of the video, link is provided, answer the question at the end.
Where,
t = time,
T(t) = temperature of the given body at time t,
Ts = surrounding temperature,
To = initial temperature of the body,
k = constant.
EXAMPLE: Using Newton's law of cooling - Bing video
(To open the video: Put your cursor on the blue line and press Ctrl then click on the blue line)
Question: A body at temperature 40ºC is kept in a surrounding of constant temperature 20ºC. It is observed that its temperature falls to 35ºC in 10 minutes. Find how much more time will it take for the body to attain a temperature of 30ºC.