Calculus Answers

use the definition of the derivative to evaluate v=4/2pir^3

Lecture Notes of Athithan S

18MAB101T-Assignment-U2&3 S. Athithan

18MAB101T-Assignment-U2&3

Register No. :

Name :

Subject : 18MAB101T-Assignment-U2&3

Class/Room No. : Online/Online

Sem./Branch : I/CSE

Instructions:

Answer all the questions. Do the assignment neatly in the A4 sheets either writing or

typing and then scan it then submit to me through the Google Classroom. To follow the

uniformity you can print the above details in the front page of your assignment.

Questions to solve:

Unit-2

1. Find the total differential coefficient of the function u = tan (3x − y) + 5y+z

2. Find

dy

dx

if 4x

2 + y

3 + 5x

2y

5 = 7x

4y

2 − 2x + 9y − 3

Find the extrema of

3x^2-y^2+x^3

A resort owner wants to enclose a beachfront area for swimming activities. Based on her plan, only 3 sides will be fence with 270 meter rope and floats, while the shoreline part will be open. Determine the dimension of the 3 sides of the rectangle that will give a maximum area.

Evaluate

int(9/4) (dx/✓x)

The n-th harmonic number is defined by

Hn=∑n, i=1 (1/i) =1+1/2+1/3+…+1/n.

Show that

Hn=∫(1>0) 1−x^n/1−x dx.

Integrals are not always easy to evaluate; sometimes we need to be clever! In this problem we study the definite integral

I=∫(π>0) xsin(x)/1+cos^2(x) dx.

At first, it appears difficult to evaluate this integral with substitution (you can try with simple substitution ideas...). But it can be done! Let's see how it goes.

(a) Use the substitution u=π−x to show that

I= π/2∫(π>0)sin(x)/1+cos^2(x) dx.

(Recall that sin(π−x)=sin(x) and cos(π−x)=−cos(x).)

(b) Evaluate the remaining definite integral using substitution to get:

I=π^2/4.

Find the line y=mx through the origin, with positive slope, which together with the parabola y=x^2 encloses a region of area 4/3.

Determine whether the following series converge, converge absolutely, converge conditionally, or diverge.

1. "\\displaystyle\\sum_{k=1}^\t\u221e" 3^k/k²*2^k
2. "\\displaystyle\\sum_{k=1}^\t\u221e" (-1)^k+1 / ( √(k+1) -√k)
3. "\\displaystyle\\sum_{k=1}^\t\u221e" (-1)^k+1 [√(k+1) -√k)]