I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

## Lectures

We finished off the sections on the closed interval method, the first derivative test and the second derivative test. This means that we can find the

• absolute maxima/minima of continuous functions defined on closed intervals
• local maxima/ minima of continuous functions defined on the entire real line
• local maxima/ minima of some differentiable functions defined on the entire real line

This means that we have enough theory to do the applied optimisation problems. However as we have only two lectures left and I don’t want to rush anything we will not be doing these in class. These applied optimisation problems occur when we have a cost function of several variables which we want to maximise or minimise.

$C=C(x_1,x_2,\dots,x_n).$

We have only studied functions of a single variable in MS2001. However if there are relationships/constraints between the variables:

$f_i(x_1,x_2,\dots,x_n)=0$ for $i=1,2,\dots,N$,

then we may be able to eliminate all but one of the variables and write

$C=C(x)$ only.

Then we can use the methods above to find the extrema of $y=C(x)$.

As this material will not be covered in class by examples, the exam question will either ask to find the maximum of a function of  a single variable (Autumn ’12 $\ell(x)=\sqrt{a^2+x^2}$ with $a$ a constant) or the relationship between the variables will be obvious (Example on p.115, $d(x,y)=\sqrt{(x-3)^2+y^2}$ with $y=x^2$).

## Tutorials

Remember you can ask whatever you want in tutorials. If you have questions about the test or past exam papers work away.

### Tutorial 10 Question Bank

Question 4: 4(b) and (c) from MS2001: Problems (after page 102 in the notes).

Examples 2 & 3 from p.99 & 100

### Tutorial 9 Question Bank

Questions 14 – 15 from Exercise Sheet 3. Questions 1 – 2 from Exercise Sheet 4 (When asked to find the critical points of a function defined on the entire real line (rather than just on a closed interval $[a,b]$), the ‘endpoints’, $\pm\infty$ are not considered critical points.).

Question 4: 1-3, 4(a) from MS2001: Problems (after page 102 in the notes).

Questions 5, 6, 13 – 18, 28, 33  from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 8 Question Bank

Questions 4, 21, 30 and 31 are very good questions from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes) to look at for Rolle’s Theorem and the Mean Value Theorem.

Questions 6 and 10 – 12 from Exercise Sheet 3.

Also Questions 19, 32 and 36  from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 7 Question Bank

Questions 3, 5, 7, 8, 13, 16, 17 from Exercise Sheet 3.

Question 3 from MS2001: Problems (after page 102 in the notes)

Questions 14, 20, 29, 35  from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 6 Question Bank

Questions 1, 2, 3 (i), (iii), 4, 7 (i), 8 (a) (i), (ii), 9, 13 (i) from Exercise Sheet 3.

Questions 35  from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 5 Question Bank

Questions 6 (iv), (v), (vii), 8, 9  from Exercise Sheet 2 — but don’t worry about removable and essential discontinuities (after page 62 in the notes).

Question 1 from MS2001: Problems (after page 108 in the notes)

Questions 1, 2, 3, 10, 11, 12, 13, 27  from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 4 Question Bank

Question 10 from Exercise Sheet 1 (after page 59 in the notes)

Questions 5, 6 (i), (ii), (iii), 7  from Exercise Sheet 2 — but don’t worry about removable and essential discontinuities (we are using the terms skip-discontinuity and blow-up) (after page 62 in the notes)

Question 1 from MS2001: Exercises (before page 63 in the notes)

Questions 23, 31- 37 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

### Tutorial 3 Question Bank

Question 9 from Exercise Sheet 1 (after page 59 in the notes)

Questions 1 – 4 from Exercise Sheet 2 (after page 62 in the notes)

Questions 4 from MS2001: Exercises (before page 63 in the notes)

Questions 27 – 30 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

### Tutorial 2 Question Bank

Questions 4, and 6 – 8 from Exercise Sheet 1 (after page 59 in the notes).

Questions 1 – 3 from MS2001: Exercises (before page 63 in the notes)

Questions 17 – 22 and 24 – 26 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

### Tutorial 1 Question Bank

Questions 1, 2 and 5 from Exercise Sheet 1 (after page 59 in the notes).

Questions 1 to 16 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).