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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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