MAT101 Calculus I - Assignment 1 - Solutions

MAT 101 - Calculus I - Assignment 2

Please start on the Exercises from Sections 1.6 to 2.4 right away. Your first midterm on Sep 14 will be up to Sec 2.4.

Assignment 2

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Part A

The following exercises from Stewart's Essential Calculus are to be solved for presentation and discussion in the tutorials:

• [Section 1.6] 1, 3, 13, 26, 41, 47
• [Section 2.1] 3, 7, 10, 16, 26, 31, 47
• [Section 2.2] 1, 3, 6, 9, 21, 33, 43
• [Section 2.3] 10, 18, 21, 22, 33, 57, 63
• [Section 2.4] 6 to 10, 25, 27, 43, 48, 54
• [Section 2.5] 4, 23, 47, 55, 65, 66
• [Section 2.6] 8, 16, 19, 32
• [Section 2.7] 2, 9, 13, 24
• [Section 2.8] 7, 21
• [Section 3.1] 13, 23, 24
• [Section 3.2] 22, 32, 35, 66
• [Section 3.3] 7, 29, 39, 58
• [Section 3.4] 9, 20

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Part B

Submit written solutions to the following to your tutor by September 23:

• [Section 1.6] 19, 23, 49
• [Section 2.1] 28, 48
• [Section 2.2] 4, 23, 39
• [Section 2.3] 23, 40, 67
• [Section 2.4] 16, 30, 51
• [Section 2.5] 49, 57, 69
• [Section 2.6] 11, 17, 44
• [Section 2.7] 11, 28, 38
• [Section 2.8] 13, 24
• [Section 3.1] 25, 31
• [Section 3.2] 23, 38, 78
• [Section 3.3] 23, 62, 68
• [Section 3.4] 17

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Extra Credit

Submit the following to Prof Habib by September 16: Consider the cubic function \(f(x)=x^3+ax^2+bx+c\). Calculate \(\lim\limits_{x\to\pm\infty}\frac{f(x)}{x^3}\)and use this to show that there is a real number \(c\) such that \(f(c)=0\).

Edit on Sep 16: Last date of submission of Extra Credit problem is changed to Friday, Sep  20. Submission must be to Prof Habib. Also note that there are two uses of \(c\) in the problem which is an oversight. Change the second \(c\) to \(d\).

MAT101 Calculus I - Assignment 1

Department of Mathematics, Shiv Nadar University
Monsoon Semester 2013-14
MAT 101 Calculus I

Assignment 1

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Part A

The following exercises from Stewart's Essential Calculus are to be solved for presentation & discussion in the tutorials:

• [Section 1.1] 1, 3, 4, 17, 23, 25, 28, 35, 43, 51, 57, 59, 61
• [Section 1.2] 2, 4, 5, 8, 13, 17, 23, 26, 35, 52, 53
• [Section 1.3] 3, 7, 11, 23, 29
• [Section 1.4] 10, 13, 21, 22, 30, 43, 46
• [Section 1.5] 3, 10, 13, 15, 26, 35, 37

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Part B

Submit written solutions of the following to your Tutor by September 2:

• [Section 1.1] 5, 6, 18, 44, 62
• [Section 1.2] 18, 49, 58, 62
• [Section 1.3] 9, 27, 43
• [Section 1.4] 7, 23, 31
• [Section 1.5] 16, 31, 40

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Extra Credit

• [Section 1.5] 47

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Precalculus Assignment 2

SNU-Precalculus-2011-Assgt2

ODE Assignment 1 Solutions

Solutions to some of the ODE first assignment problems. Scroll through the embedded document using the controls at its base.

Assignment 1

ODE - Assignment 1

Due Date: September 9 September 12

1. A tank contains 800 gal of water in which 200 lb of salt is dissolved. Two gallons of fresh water runs in per minute and 2 gal of the mixture in the tank, kept uniform by stirring, runs out per minute.
   
   1. Form a differential equation for the amount of salt in the tank as a function of time.
   2. How much salt is left in the tank after 5 hours?
2. A thermometer, reading $10^\circ\mathrm{C}$, is brought into a room whose temperature is $23^\circ\mathrm{C}$. Two minutes later the thermometer reading is $18^\circ\mathrm{C}$.
   
   1. Use Newton's Law of Cooling to model the change of the thermometer reading with time.
   2. How long will it take until the reading is practically $23^\circ\mathrm{C}$, say, $22.8^\circ\mathrm{C}$?
3. Solve the following initial value problem (IVP) and make a reasonably accurate sketch of the solution:
   \[ 2y^\prime + y^3 = 0;\quad y(0)=1 \]
   
4. Solve the following ODEs:
   
   1. $\left[\sin(y)\cos(y)+x\cos^2(y)\right]\,dx + x\,dy=0$
   2. $e^y\,\left[\sinh(x)\,dx + \cosh(x)\,dy\right]=0$
   3. $(x^2+y^2)\,dx - 2xy\,dy =0$
   4. $\left(\cos(xy)+\dfrac{x}{y}\right)\,dx + \left(1 + \dfrac{x}{y}\cos(xy)\right)\,dy=0$
5. 1. Under what conditions for the constants $A$,$B$,$C$,$D$, is the following ODE exact?
      \[(Ax+By)\,dx + (Cx+Dy)\,dy=0\]
   2. Solve this exact ODE.
6. 1. Solve the ordinary differential equation: $y^\prime\tan(x)=2y-8$.
   2. Sketch the family of solutions given by the general solution to the above ODE.
   3. Give the particular solution to the above ODE such that $y=0$ when $x=\pi/2$.

Precalculus - Assignment 1

Submit by: September 12

(Wait a few seconds for the images of the formulas to load)

1. Let $f:A\to B$ and $g:B\to A$ such that $g\circ f =\mathrm{id}_A$. State whether the following conclusions are valid. For valid conclusions give a proof. For invalid conclusions give a counterexample.
   
   1. $f$ is a bijection.
   2. $f$ is onto.
   3. $f$ is one-one.
   4. $g$ is a bijection.
   5. $g$ is onto.
   6. $g$ is one-one.
2. Let $A$ and $B$ be sets with 3 and 4 elements respectively. Can there be a bijection between these sets?
   
3. Exhibit a bijection from the set of even natural numbers $2\mathbb N$ to the set of all natural numbers $\mathbb N$.
   
4. Exhibit a bijection from $\mathbb N$ to the set of integers $\mathbb Z$.
   
5. Are the following functions one-one or onto?
   
   1. $f:[0,1]\to[a,b]$, $f(x)=bx+(1-x)a$.
   2. $f:\mathbb R\to\mathbb R$, $f(x)=x^2+x+1$.
   3. $f:\mathbb R\to\mathbb R$, $f(x)=x+|x|$.
6. Let $f:\mathbb N\to A$ and $g:\mathbb N\to B$ be surjective. Show there is a surjective map $h:\mathbb N\to A\cup B$.
   
7. Let $f:\mathbb R^2\to\mathbb R$ be defined by $f(x,y)=xy$. What are $f^{-1}(r)$ for $r\in\mathbb R$ and $f^{-1}([a,b])$? Draw pictures of these inverse images.
   
8. Let $f:X\to Y$. Show that
   
   1. $f$ is onto iff $f(f^{-1}(B))=B$ for every $B\subset Y$.
   2. $f$ is one-one iff $f^{-1}(f(A))=A$ for every $A\subset X$.