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Calculus

Curve Sketching

Concept Quizzes

Curve Sketching Warmup
Turning Points
Curve Sketching
Increasing / Decreasing Functions
Vertical Asymptotes
Relative Magnitude of Functions
Tangent to a Curve
Normal to a Curve

Challenge Quizzes

Curve Sketching: Level 3 Challenges
Curve Sketching: Level 4 Challenges

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 $f'(a)?$

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by Brilliant Staff

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

\displaystyle{ y= \frac{1}{6}x-\frac{\pi}{24} }

\displaystyle{ y= {6}x-\frac{\pi}{24} }

\displaystyle{ y= -{12}x+\frac{\pi}{48}+{10} }

\displaystyle{ y= -\frac{1}{12}x+\frac{\pi}{48}+{10} }

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by Brilliant Staff

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 $(9,4)?$

\displaystyle{y= {7}x+{40} }

\displaystyle{y= {-6}x+{32} }

\displaystyle{y={9}x+{34} }

\displaystyle{y={-8}x-{37} }

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by Brilliant Staff

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

\displaystyle{y= -{9}x-\frac{17}{3} }

\displaystyle{y= \frac{1}{9}x+\frac{17}{3} }

\displaystyle{y= {9}x+\frac{17}{3} }

\displaystyle{y= -\frac{1}{9}x-\frac{17}{3} }

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by Brilliant Staff

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

\displaystyle{ y=-\frac{1}{{10}{e}^3+{6}}x+\frac{3}{{10}{e}^3+6}+{10}{e}^3+{14} }

\displaystyle{ y=\frac{1}{{10}{e}^3}x-\frac{3}{{4}{e}^3}-{8} }

\displaystyle{ y=-({10}{e}^2+{3})x+{10}{e}^3+{10}{e}^3+{14} }

\displaystyle{ y={10}{e}^3x+{4}{e}^3+{12}{e}^2+{7} }

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by Brilliant Staff