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If the tangent and normal at any point \(P\) of a conic meet the transverse axis of \(T\) and \(G\), respectively, and if \(S\) be the focus then prove that \(\frac{1}{S G}-\frac{1}{S T}\) is a constant.

Here are the results of a survey that examines preference for peanut or plain M\&Ms and exercise level. Are these variables related? Do the calculations by hand or using SPSS. \begin{tabular}{|l|l|c|c|c|} \hline & & \multicolumn{3}{|c|}{ Exercise Level } \\ \hline & & High & Medium & Low \\ \hline M\&M preference & Plain & 160 & 400 & 175 \\ \hline & Peanut & 150 & 500 & 250 \\ \hline \end{tabular}

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find an equation for the tangent line to \(f\) at the specified $$ \text { point }\left(x_{0}, f\left(x_{0}\right)\right) $$ d. Find an equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right) .\) Discuss the symmetries you see across the main diagonal (the line \(y=x\) ). $$y=\sin x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \quad x_{0}=1$$

$$ \text { 脺berpr眉fe die folgenden Matrizen auf Definitheit: } $$ $$ \left(\begin{array}{rrr} 1 & 2 & -2 \\ 2 & 2 & 0 \\ -2 & 0 & -4 \end{array}\right), \quad\left(\begin{array}{rrr} -3 & 1 & -3 \\ 1 & -2 & 0 \\ -3 & 0 & -4 \end{array}\right),\left(\begin{array}{rrr} 7 & 0 & -8 \\ 0 & 1 & 2 \\ -8 & 2 & 17 \end{array}\right) $$

Which of the following are APs ? If they form an AP, find the common difference \(d\) and write three more terms. (i) \(2,4,8,16, \ldots\) (ii) \(2, \frac{5}{2}, 3, \frac{7}{2}, \ldots\) (iii) \(-1.2,-3.2,-5.2,-7.2, \ldots\) (iv) \(-10,-6,-2,2, \ldots\) (v) \(3,3+\sqrt{2}, 3+2 \sqrt{2}, 3+3 \sqrt{2}, \ldots\) (vi) \(0.2,0.22,0.222,0.2222, \ldots\) (vii) \(0,-4,-8,-12, \ldots\) (viii) \(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}, \ldots\) (ix) \(1,3,9,27, \ldots\) (x) \(a, 2 a, 3 a, 4 a, \ldots\) (xi) \(a, a^{2}, a^{3}, a^{4}, \ldots\) (xii) \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots\) (xiii) \(\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots\) (xiv) \(1^{2}, 3^{2}, 5^{2}, 7^{2}, \ldots\) (xv) \(1^{2}, 5^{2}, 7^{2}, 73, \ldots\)

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