8.6 Quiz 3

Please submit solutions to the following via moodle by 23:59 on Wednesday 7th December. Each question is worth [2] marks. For Q3.4-Q3.5, please see Week08Looping.pdf.

Q3.1Length of catenary. Let $C$ be the curve $y=\cosh x$ for $-a<x<a,$ which is a catenary. The length $L$ of $C$ is given by which of the following?

(A) $L={{1}\over{2}}\sinh 2a+a$. $\;\;\;\;$ (B) $L={{1}\over{4}}\sinh 2a+{{a}\over{2}}.$ $\;\;\;\;$ (C) $L=\cosh a.$

(D) $L=2\sinh a$.$\;\;\;\;$     (E) $L=\int_{-a}^{a}\sqrt{1+\sin^{2}x}\,dx.$

Q3.2Tangents. Let $C$ be the curve that is given implicitly by the equation

Which of the following statements about the tangent line to $C$ at $P=(1,-2)$ is true?

(A) The equation of the tangent line to $C$ at $P$ is $y=-\frac{9}{4}x$.

(B) The tangent line crosses the $x$-axis at $(\frac{1}{9},0)$.

(C) The tangent line passes through the point $(-1,1)$.

(D) The point $(1,-2)$ is not on $C$, so there is no tangent to $C$ at $P$.

(E) The normal line to $C$ at $P$ has gradient $-{9}$.

Q3.3Stationary points. How many stationary points does the function $f(x,y)=\frac{x^{3}}{3}-xy^{2}+\frac{2y^{5}}{5}$ have?

1. (A)

   None.

2. (B)

   1.

3. (C)

   2.

4. (D)

   3.

5. (E)

   Infinitely many.

Q3.4Recursion.Construct a for loop that performs the following recursion:

Correct to 2 decimal places, what is ${\color{blue}\colorlet{pgfstrokecolor}{.}x_{7}}$ when the initial value is ${\color{blue}\colorlet{pgfstrokecolor}{.}x_{0}=\sqrt{\pi}}$?

Q3.5Nested For Loops. Suppose I have the following numbers of coins of different denominations:

What is the number of distinct ways in which I could make exactly £1 from subsets of those coins?

[For instance ($5\times$ 2p, $2\times$ 5p and $4\times$ 20p) would be one possible way.]

1. (A)

   52

2. (B)

   72

3. (C)

   82

4. (D)

   92

5. (E)

   102