Answer To: Assignment 1 Complete this assignment after you have finished Unit 3, and submit your work to your...
David answered on Dec 22 2021
1. Indicate whether each of the following functions is invertible in the given interval. Explain
a. [ )
The given function is a continuously decreasing function. Any line drawn parallel to x – axis
cuts the graph of the function at a single point. Hence we can say that the given function is
invertible.
b. ( ) ( ]
The given function is a oscillating. Any line drawn parallel to x – axis cuts the graph of the
function at more than a single point. Hence we can say that the given function is not
invertible
.
c.
( ]
The given function is decreases and increases. Lines drawn parallel to x – axis (near y=1) cuts
the graph of the function at more than a single point. Hence we can say that the given
function is not invertible.
2.
a. Use Definition 1 on page 38 of the textbook to show that the function is one – one
( )
Sol:
Let x1, x2 be such that,
( ) ( )
(
)
(
)
(
)
(
)
Hence the given function is one –one.
b. Find the inverse function, f-1(x)
Sol:
Let the f-1(x) =y
( )
(
( )
)
( )
c. Give the domain and range of the functions f and f-1.
Sol:
The functions f and f-1 cannot take the values x=5/2 and x=-3/2 respectively as these
values make the denominators 0, which in turn makes the value of the function undefined.
Hence the domain of the function f is ,
- and the domain of the function f-1 is ,
-
The range of function f cannot have the value that is not in the domain of the function f-1 and
vice – versa.
Hence the range of the function f is ,
- and the range of the function f-1 is ,
-
3. Find the solution of the following equations, if the solution exists.
a.
Sol:
The equation can be reduced to,
Taking logarithm on both sides and solving the quadratic equation obtained, we get,
b. ( ) ( )
Sol:
Using the property of the logarithm that ln(a)+ln(b)=ln(ab), we can write,
(( )( )) ( )
( )
Solving the quadratic equation we get the roots to be,
√
c. ( )
Sol:
Taking cos on both sides, we get,
d. ( )
Sol:
Taking sec on both sides, we get,
( )
( )
4. Use logarithmic differentiation to find,
a.
Sol:
Let
( )
( ( )) ( ) ( ) ( )
( )
( )
( ) (√ )
( ) (√ )
( )
( ) (√ )
( ) (√ )
( ) ( )
( )
b.
Sol:
Let ( )
We know that, ( √ ) ( )
Differentiating this, we get,
√
In this case, y=e2x.
Hence,
( )
√
( )
5. Evaluate the following limits, justifying your answers. If a limit does not exist, explain why.
a.
This function is taking the form
. No matter the values of x, the value of cosx and sinx
remain bounded in [-1, 1].
Hence we can say that, both cosx ,sinx divided by x, as x approaches infinity, approach 0.
Divide numerator and denominator with x3.
b.
(
)
Assuming , this can be...