heureka

1. The magnitude of  \(\vec{u}\).

Round your answer to at least four decimal places.

\(\vec{u} = \binom{-8}{7}\)

\(\begin{array}{|rcll|} \hline ||\vec{v}|| &=& \sqrt{(-8)^2+7^2} \\ &=& \sqrt{64+49} \\ &=& \sqrt{113} \\ &=& 10.6301458127 \\ &\approx& 10.6301 \\ \hline \end{array}\)

2.The direction of \(\vec{u}\), thaat is the angle \(\theta\) it makes with the positive x-axis.

State your answer in degrees, rounded to at least four decimal places.

\(\vec{u} = \binom{-8}{7} \)

\(\begin{array}{|rcll|} \hline \tan(\theta) &=& \frac{|~\vec{e_x} \times \vec{u}~| } {\vec{e_x} \cdot \vec{v} } \\ &=& \frac{ \left|~\binom{1}{0} \times \binom{-8}{7}~\right| } {\binom{1}{0} \cdot \binom{-8}{7} } \\ &=& \frac{ (1)\cdot (7) - (0)\cdot (-8) } { (1)\cdot (-8) + (0)\cdot (7) } \\ &=& \frac{ 7+0 } { -8+0 } \\ &=& \frac{ 7} { -8 } \quad & | \quad II.\text{Quadrant} \\ \theta &=& \arctan(\frac{ 7 } { -8 }) \\ \theta &=& \arctan(-0.875) \\ \theta &=& -41.1859251657^{\circ} + 180^{\circ} \quad & | \quad II.\text{Quadrant} \\ \theta &=& 138.814074834^{\circ} \\ \theta &\approx& 138.8141^{\circ} \\ \hline \end{array} \)

Find the angle between the vectors. State your answer in degrees,

rounded to at least four decimal places.

\(\vec{u} = \binom{-2}{8} \\ \vec{v} = \binom{-7}{-9} \\\)

\(\begin{array}{|rcll|} \hline \tan(\theta) &=& \frac{|~\vec{u} \times \vec{v}~| } {\vec{u} \cdot \vec{v} } \\ &=& \frac{ \left|~\binom{-2}{8} \times \binom{-7}{-9}~\right| } {\binom{-2}{8} \cdot \binom{-7}{-9} } \\ &=& \frac{ (-2)\cdot (-9) - (8)\cdot (-7) } { (-2)\cdot (-7) + (8)\cdot (-9) } \\ &=& \frac{ 18+56 } { 14-72 } \\ &=& \frac{ 74 } { -58 } \quad & | \quad II.\text{Quadrant} \\ &=& \frac{ 37 } { -29 } \\ \theta &=& \arctan(\frac{ 37 } { -29 }) \\ \theta &=& \arctan(-1.27586206897) \\ \theta &=& -51.9112271190180^{\circ} + 180^{\circ} \quad & | \quad II.\text{Quadrant} \\ \theta &=& 128.088772881^{\circ} \\ \theta &\approx& 128.0888^{\circ} \\ \hline \end{array}\)

850e^(0.055x)=195e^(0.075x)

\(\begin{array}{|rcll|} \hline 850\cdot e^{0.055x} &=& 195\cdot e^{0.075x} \quad & | \quad : 195 \\ \frac{850}{195} \cdot e^{0.055x} &=& e^{0.075x} \quad & | \quad \cdot e^{-0.055x} \\ \frac{850}{195} &=& e^{0.075x} \cdot e^{-0.055x} \\ \frac{850}{195} &=& e^{0.075x-0.055x} \\ \frac{850}{195} &=& e^{0.02x} \\ \frac{170}{39} &=& e^{0.02x} \quad & | \quad \text{ln of both sides } \\ \ln(\frac{170}{39}) &=& \ln(e^{0.02x}) \\ \ln(\frac{170}{39}) &=& 0.02x\cdot \ln(e) \quad & | \quad \ln(e) = 1 \\ \ln(\frac{170}{39}) &=& 0.02x \quad & | \quad : 0.02 \\ \frac{ \ln(\frac{170}{39}) } {0.02} &=& x \\ \frac{ \ln(4.35897435897) } {0.02} &=& x \\ \frac{ 1.47223679092 } {0.02} &=& x \\ 73.6118395460 &=& x \\ \hline \end{array}\)

Convert the base-64 number

\(100_{64}\) 

to base 62.

\(\begin{array}{lcll} 100_{64} \text{ to base }10\\ \hline (1\cdot 64+0)\cdot 64 + 0 &=& 64^2 \\ &=& 4096_{10} \\ \hline \end{array}\)

\(100_{64} = 4096_{10}\)

\(\begin{array}{lcll} 4096_{10}\text{ to base } 62 \\ \hline 4096 &:& 62 &=& 66\cdot 62 &+& 4 \\ 66 &:& 62 &=& 1\cdot 62 &+& 4 \\ 1 &:& 62 &=& 0\cdot 62 &+& 1 \\ \hline \end{array} \)

\(100_{64} = 4096_{10} = 144_{62}\)

Find the area under the standard normal curve between z=-2.88 and z=1.95.

z=1.95

Cumulative area: 0.97441194

z=-2.88

Cumulative area: 0.00198838

0.97441194 - 0.00198838 = 0.97242356

The area under the standard normal curve between z=-2.88 and z=1.95 is 0.97242356

The equation noted by Gauss should be of course:

\(\begin{array}{|rcll|} \hline A_2(x) &=& 1 + 4\cdot \left[~ \sqrt x ~\right] + 4 \cdot \left[~ \sqrt{\frac{x}{2}} ~\right]^2 + \ 8 \sum \limits_{y_1=\left[~ \sqrt{\frac{x}{2}} ~\right]+1}^{\left[~ \sqrt x ~\right]} \left[~ \sqrt{x-y_1^2} ~\right]\\ \hline \end{array}\)

Calculation of the Number of Lattice Points in the Circle

Let \(r\) be the Radius of the Circle

Let \(x = r^2\)

Let \( A_2(x)\) be the number of lattice points in circle

Let \([\ldots] \) be the largest integer

Noted by Gauss:

\(\begin{array}{|rcll|} \hline A_2(x) &=& 1 + 4\cdot \left[~ \sqrt x ~\right] + 4 \cdot \left[~ \sqrt{\frac{x}{2}} ~\right]^2 + \sum \limits_{y_1=\left[~ \sqrt{\frac{x}{2}} ~\right]+1}^{\left[~ \sqrt x ~\right]} \left[~ \sqrt{x-y_1^2} ~\right]\\ \hline \end{array}\)

Computed Results

\(\begin{array}{r|r} \hline r = \sqrt{x} & A_2(x) \\ \hline 1&5\\ 2&13\\ 3&29\\ 4&49\\ 5&81\\ 6&113\\ 7&149\\ 8&197\\ 9&253\\ 10&317\\ 11&377\\ 12&441\\ 13&529\\ 14&613\\ 15&709\\ 16&797\\ 17&901\\ 18&1009\\ 19&1129\\ 20&1257\\ 21&1373\\ 22&1517\\ 23&1653\\ 24&1793\\ 25&1961\\ 26&2121\\ 27&2289\\ 28&2453\\ 29&2629\\ 30&2821\\ 31&3001\\ 32&3209\\ 33&3409\\ 34&3625\\ 35&3853\\ 36&4053\\ 37&4293\\ 38&4513\\ 39&4777\\ 40&5025\\ 41&5261\\ 42&5525\\ 43&5789\\ 44&6077\\ 45&6361\\ 46&6625\\ 47&6921\\ 48&7213\\ 49&7525\\ \hline \end{array}\)

Continued

\(\begin{array}{r|r} \hline r = \sqrt{x} & A_2(x) \\ \hline 50&7845\\ 51&8173\\ 52&8497\\ 53&8809\\ 54&9145\\ 55&9477\\ 56&9845\\ 57&10189\\ 58&10557\\ 59&10913\\ 60&11289\\ 61&11681\\ 62&12061\\ 63&12453\\ 64&12853\\ 65&13273\\ 66&13673\\ 67&14073\\ 68&14505\\ 69&14949\\ 70&15373\\ 71&15813\\ 72&16241\\ 73&16729\\ 74&17193\\ 75&17665\\ 76&18125\\ 77&18605\\ 78&19109\\ 79&19577\\ 80&20081\\ 81&20593\\ 82&21101\\ 83&21629\\ 84&22133\\ 85&22701\\ 86&23217\\ 87&23769\\ 88&24313\\ 89&24845\\ 90&25445\\ 91&25997\\ 92&26565\\ 93&27145\\ 94&27729\\ 95&28345\\ 96&28917\\ 97&29525\\ 98&30149\\ 99&30757\\ 100&31417\\ \hline \end{array}\)

ʃ (2x + 6)(x2 + 6x + 3)7 dx

Sorry, without mistakes:

\(\begin{array}{rcll} \hline && \int (2x + 6)(x^2 + 6x + 3)\cdot 7\ dx \\ \hline \end{array}\)

\(\begin{array}{rcll} &=& 7\cdot \int (2x + 6)(x^2 + 6x + 3)\ dx \\ &=& 7\cdot \int (2x^3+12x^2+6x+6x^2+36x+18)\ dx \\ &=& 7\cdot \int (2x^3+18x^2+42x+18)\ dx \\ &=& 7\cdot \left( \frac24x^4+\frac{18}{3}x^3+\frac{42}{2}x^2 + 18x \right) \\ &=& 3.5x^4+42x^3+147x^2+126x + c \\ \end{array}\)

3+2+1+2+5+2+2+3+12+2+3+2+3+4+5+3+3+5+6+3+2+1+5+3+2+2+3+3+1+5+3+2+22+2+2+3+4+5+3+2+5+3+3+1+5+2+3+2 = 170

ʃ (2x + 6)(x2 + 6x + 3)7 dx

\(\begin{array}{|rcll|} \hline && \int (2x + 6)(x^2 + 6x + 3)\cdot 7\ dx \\ &=& 7\cdot \int (2x + 6)(x^2 + 6x + 3)\ dx \\ &=& 7\cdot \int (2x^3+12x^2+6x+6x^2+36x+18)\ dx \\ &=& 7\cdot \int (2x^3+18x^2+42x+18)\ dx \\ &=& 7\cdot \left( \frac24x^4+\frac{18}{3}x^3+\frac{42}{2}x^2 + 18x \right) \\ &=& 3.5x^4+42x^3+147x^2+126 + c \\ \hline \end{array} \)