Question 1
Differentiating
(a). f(x)=6x^2ln(4x)
f’(x)=d/dx(6x^2ln(4x))
First we remove the constant
f’(x)=6 d/dx(x^2ln(4x))
Second the product rule
(f*g)’=f’*g+f*g’
f=x^2 , g=ln(4x)
f’=d/dx(x^2)=2x , g’=d/dx(ln(4x))=1/x
Third substitute the values into the product rule
f’(x)=6 (2xln(4x)+(1/x)*x^2)
Final Result
f’(x)=6(2xln(4x)+x)
(b). f(x)=
f’(x)=d/dx()
First use the quotient rule
(f/g)’=(f’*g-f*g’)/g^2
f=e^-2x-1 , g=cos(2x)
f’=2e^-2x , g’=-2sin(2x)
Second substitute the values in the quotient rule
f’(x)=
Final Result
f’(x)=
(c). h(t)=
h’(t)=
First use the chain rule
dfh(u)/dt=dh/du*du/dt
h= , u=)
d/du(=1/2
d/dt())=
Second substitute the values in the chain rule
h’(t)=1/2 *)
h’(t)=1/(2)*)
Final Result
h’(t)=
Question 2
Closed box
(a). V=L*W*H
V=2x*x*h
V=
Expression of h in terms of x if V=1000
h=1000/
(b). A=2LW+2LH+2WH
A=2(2x*x)+2(2x*h)+2(x*h)
Final result
A=
A=
A=
A=
(c). A’=
A’=
0=
x=7.211248
A”=8+6000/x^3
A”=24
h=1000/2x^2
h=9.615
Question 3
Expansion of the expression =
Using the Sum rule
= –
==-cos(x)
==x
Final result
=-cos(x)-x+C
Question 4
(a).
First using substitution
U=cos(x)
=
By taking out the constant
=
Applying substitution with V=3u-4
=
Taking the constant out
=
=
Applying the power rule
=
Substituting back all the values
V=3u-4 and u=cos(x)
=
Final result
=
(b).
Taking the constant out
=
Making u=2x^4 and substituting
=
=
=
We end up with
(
=1.59726
Final Answer
1.597
Question 5
Using integration by parts
Let u= and V=sin(x)
V’=-cos(x) , u’=2(x+3)
Hence
=
Further integrating the second part
=
Hence
Simplifying
=
Final result
=+C
Question 6
Area under a curve
(a). The value of f(x) for both x’s
f(x)=
f(x)=2.021
f(x)=
f(x)=2.56
Hence, the region has positive value meaning it is above the x-axis
(b) Upper limit (-Lower limit (
Where the limits are the given values for upper and lower bounds
(c).
Integration by part
Where
U=(x+1), V=sin (5x)
U’=1, V’=-1/5cos(5x)
Hence
=
Integrating the second part
=
Finally
=+C
After integration
) for =
) for =
=
=
Area=-2.285
Question 8
(a)
(i) AB
(ii). BA
Not possible, the number of columns in the first matrix must be equivalent to the number of row of the second matrix
(iii). A^2
No possible, a matrix can only be raised to a given power if the matrix has the same number of rows and columns
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