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Curve Sketching

You don't need a calculator or computer to draw your graphs! Derivatives and other Calculus techniques give direct insights into the geometric behavior of curves.

Level 3


Let \(f(x) = x^3 + ax^2 + bx + c\), where \(a, b\), and \(c\) are real numbers. In order for \(f(x)\) to be invertible, \(a\) and \(b\) must be related as: \(\dfrac{a^m}{b^n} \leq p \), where \(m, n\), and \(p\) are also real numbers.

Find the mimimum value of \(m + n + p\).

What is the acute angle between the curves \(y=\sin x\) and \(y=\cos x\)?

Note: Angle between two curves is the angle between the tangent lines at the point of intersection of the two curves.

A tangent to the function \(f(x)=\dfrac{1}{x}\) is drawn on the first quadrant of the Cartesian plane.

Let \(X\) denote the area bounded by this tangent, the \(x\)-axis and the \(y\)-axis.

What is the range of the values of \(X\) can take?

\[6 \ln (x^2 + 1) - x = 0 \]

How many real solutions exist for the equation above?

Which of the given options does the graph represent?

\[\quad \begin{matrix} (a)\quad \left| y \right| =\cos { x } \quad & (b)\quad \left| y \right| =\sin { x\quad \quad } \\ (c)\quad y=\left| \cos { x } \right| \quad & (d)\quad \left| y \right| =\left| \cos { x } \right| \end{matrix}\]


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