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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.

Normal to a Curve

Below is the graph of the curve \(y=f(x).\) The green line is tangential to the curve at the point \(P=(a,f(a)).\) If the red line is perpendicular to the green line and the equation of the red line is \(y=-\frac{1}{5}x+15,\) what is the value of \(f'(a)?\)

What is the normal to the curve \( y=6\tan x + 4 \) at the point \( \left(\frac{\pi}{4}, 10\right) \)?

What is the equation of the line \(l\) which passes through the point \((5,2)\) and is perpendicular to the line passing through the two points \( (3,3)\) and \( (9,4)? \)

What is the normal to the curve \( y=2x^2 - 3x -15 \) at the point \( (3, -6) \)?

What is the normal to the curve \( y=10{e}^x + {x}^2 +5 \) at the point \( (3,10{e}^3+14 ) \)?

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