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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 4

         

How many real values of \(x\) satisfy the equation

\[\large {x}^{2}-{2}^{x}=0? \]

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

How many real solutions exist for the equation above?

On a vertical parabola \(P_1\), pick a point \(T\). Another vertical parabola \(P_2\) is tangent to \(P_1\) at \(T\). Spanning through all possible parabolas \(P_2\), what is the locus of the focus of \(P_2\)?

Let \(f(x)=x^4-6x^2+5.\) If \(P(x_0,y_0)\) is a point such that \(y_0>f(x_0)\) and there are exactly two distinct tangents drawn to the curve \(y=f(x),\) what is the maximum value of \(y_0?\)

Try : Part-2

The function \(f(x)= x^{3}-11x^{2} +19x+13\) has zeros \(a_{1},a_{2},a_{3}.\) Then what is the value of \([ a_{1} ]+[ a_{2} ]+[ a_{3} ]?\)

Note: \([m]\) Represents the greatest integer less than or equal to \(m.\)

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