Mathematics Assignment

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Added on  2023/06/04

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This Mathematics Assignment includes solutions to questions on radioactive decay, finding the smallest value of a function, sketching graphs and finding inverse functions. The subject is Mathematics and the course code and college/university are not mentioned.

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Mathematics Assignment
Mathematics Assignment
Student’s Name
Institution Affiliation
Calculus

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Mathematics Assignment
Question one
Radioactive Decay
The amount of decaying substance at time t is given by
A=A0 ekt , ..i
Given information
when t=1 , A=60.7when t=6 , A=5
To obtain the value of A0 k , values oft A, will be substituted in the equation i to
obtain two equations.
60.7=A0 ek ii
5=A0 e6 k . iii
Introduce natural logarithm
ln 60.7=ln ( A0 ek )
4.1059=ln A0 +k ii
ln 5=ln ( A ¿¿ 0 e6 k) ¿
1.6094=ln A0 +6 k iii
Below the equation obtained
4.1059=ln A0 + k ii
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Mathematics Assignment
1.6094=ln A0 +6 k iii
Solve the two by elimination method
6 ( 4.1059 ) =6 ln A0 +6 k .. ii
1.6094=ln A0 +6 k iii
Subtract the iii from the ii,
23.02622=5 l n A0
ln A0 =23.02622
5
ln A0 =4.6052
A0 =100.0075
A0 100.
Substitute A0 =100.0075, in equation iii to obtain value of k
1.6094=ln ( 100.0075 ) +6 k
1.6094=4.6052+6 k
6 k =2.9958
k =0.4993
k 0.5
Therefore, the value of A0 k are A0 100 andk 0.5 respectively.
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Mathematics Assignment
Question Two
The function f : [a , 4 ] C , f ( x ) =16x2
a. The smallest value of a such that f ( x ) is11 function
To find the smallest of a the derivative of f (c ) is equated to 0
0=f ' ( c ) = f ( b ) f ( a )
ba , where b=4a=a are intervals of C
0=16 ( 4 )2 (16 ( a2 ) )
4a
¿ 0(16a2 )
4a
¿ (1 ) ( 4a ) ( 4 +a )
4a
0= ( 1 ) ( 4+ a )
0=4 +a
a=4
b. Value of C
C=f [ a , b ]
when b=4 ,C=0when a=4 ,C=0
Therefore, is C 0
c. Rule for the inverse of f −1

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Mathematics Assignment
To do this the f (x) is change to y and then y is replaced with x and solve for y
x=16 y2
x16= y2
y=± x16
Therefore, f1= y =± x16
The domain, consideringf (x)=16x2, value of x has greater than or equal to 4
Solve 16x2 0
16 x2
4 x
Therefore,
Domain [ , 4 ] and range is [ ,0 ]
The domain of inverse function is given by the range of original function while range is
given by domain
Thus domain and range for the inverse function are [ ,0 ] and [ 4 , ] respectively.
d. Sketch graph
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Mathematics Assignment
Question three
Function to be considered: f(x) = 1 − loge(x + 1)
a. Stating the transformation that have occurred to g ( x )=loge x form f(x) = 1 loge(x +
1)
f(x) is graph of g ( x ) =loge x translated 1 unit parallel to X-axis and cut y axis
at -1
b. Coordinates where the graph cuts the x and y-axis
When y=0 and x=0
y=0
0=1loge(x +1)
1=loge ( x +1 ) , change ¿ index form
e=x +1
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Mathematics Assignment
x=1.718
When x=0
y=1loge ( 0+1 )
y=1loge ( 1 )
y=1
Therefore the graph cuts the y and x axes at point (0,1) and (1.718,0) respectively.
c. Sketch of the graph
.

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Mathematics Assignment
d. Inverse of function
f(x) = 1 − loge(x + 1)
y=1loge (x +1) Replace x with y and y with x and solve for y
x=1loge ( y +1 )
x1=loge ( y +1 )
( x1 ) =loge ( y +1 )
y +1=e (x1 )
y=e (x1 )1
Therefore, the inverse function of y=f (x ) is
e ( x1 )1
To sketch the graph, the domain and range of the inverse need to be determined.
First, determine the domain and range of y=1loge ( x +1 ), you are interested with
1loge ( x +1 ) 0
1 loge ( x+1 )
e x +1
1.718 x
Thus the domain and range will be [ 1 ,1.718 ] and [0, 1.718] respectively
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Mathematics Assignment
They will be given by reverse of the domain and range of y=f (x ), that is
Domain: [ 0 , 1.718 ]
Range;[ 1 ,1.718 ]
Below is the sketch graph, in blue
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Mathematics Assignment
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