Assignment 1 Complete this assignment after you have finished Unit 3, and submit your work to your tutor for grading. Remember to attach a tutor-marked exercise form (paper or electronic). Be certain...

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Assignment 1 Complete this assignment after you have finished Unit 3, and submit your work to your tutor for grading. Remember to attach a tutor-marked exercise form (paper or electronic). Be certain to keep a copy of your work, in case the original is lost in the mail. This exercise is worth 15 per cent of your final grade. Total points: 170 (plus 10 bonus points) 9 pts 1. Indicate whether each of the following functions is invertible in the given interval. Explain. a. sechx on [0, 8) b. cos(ln x) on (0, ep ] c. e x 2 on (-1, 2] 9 pts 2. a. Use Definition 1 on page 38 of the textbook to show that the function f(x) = 3x + 4 5 - 2x is one-to-one. b. Find the inverse function, f -1 (x). c. Give the domain and range of the functions f and f -1 . 12 pts 3. Find the solution of the following equations, if the solution exists. a. e x 2+3x+1 e x = e b. ln(5x + 6) + ln(x - 2) = 1 c. cos-1 (3x - 1) = 0 d. sec-1 (3x - 2) = 1 10 pts 4. Use logarithmic differentiation to find a. d dx sin-1 (x 2 ) sinh-1 (x 2 ) sin4 (x 2) b. d 2 dx2 sech-1 (e 2x )


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Assignment1 Complete this assignment after you have ?nished Unit 3, and submit your work to your tutor for grading. Remember to attach a tutor-marked exercise form (paper or electronic). Be certain to keep a copy of your work, in case the original is lost in the mail. This exercise is worth 15 per cent of your ?nal grade. Totalpoints: 170 (plus 10 bonus points) 9pts 1. Indicate whether each of the following functions is invertible in the given interval. Explain. a. sechx on [0;1)  b. cos(lnx) on (0;e ] 2 x c. e on (1; 2] 9pts 2. a. Use De?nition 1 on page 38 of the textbook to show that the function 3x + 4 f(x) = 5 2x is one-to-one. 1 b. Find the inverse function,f (x). 1 c. Give the domain and range of the functionsf andf . 12pts 3. Find the solution of the following equations, if the solution exists. 2 x +3x+1 e a. =e x e b. ln(5x + 6) + ln(x 2) = 1 1 c. cos (3x 1) = 0 1 d. sec (3x 2) = 1 10pts 4. Use logarithmic differentiation to ?nd 1 1 2 2 d sin (x ) sinh (x ) a. 4 2 dx sin (x ) 2 d 1 2x b. sech (e ) 2 dx IntroductiontoCalculusII16pts 5. Evaluate the following limits, justifying your answers. If a limit does not exist explain why. 3 3x + cosx a. lim 3 x!1 sinxx   1 1 tan x b. lim + x! x p c. lim x + sinx lnx + x!0 1 cos x d. lim x!1 1x 6pts 6. Give the equation of the line tangent to the curve at the given point. p 1 a. y tan x =xy at ( 3; 0) 2 x 2 b. lny =x + 2e at (0;e ) 6pts 7. Describe the strategy you would use to integrate Z m n cot x csc xdx ifm andn are odd. 40pts 8. Integrate each of the expressions below, and identify the technique of integration you apply. Z a. cothxdx Z 2 b. x cosxdx Z 1 cosh x c. p dx 2 x 1 Z d. tanh(lnx)dx Z 2 e. sin x sin(2x)dx Z 3 f. cotx csc xdx Z p 2 g. x 6xx 8dx Z x h. p dx 2 6xx Z cosx i. dx 2 sin x 2 sinx 8 Z dx j. p 3 x + x Mathematics 266 /12pts 9. Estimate Z 3 dx 3 1 +x 0 using a. the Midpoint Rule, withn = 3. b. the Trapezoidal Rule, withn = 6. c. Simpson’s Rule, withn = 6. 10pts...



Answered Same DayDec 22, 2021

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
123 Votes
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 i
s 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...
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