Question: "How do you write an equation of the..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Lubna Boyce

Answered question

2022-02-18

How do you write an equation of the line tangent to the graph of

f (x) = 7 - 15 x + 9 x^{2} - x^{3}

at x=2?

Answer & Explanation

legertopdxa

Beginner2022-02-19Added 8 answers

f (x) = 7 - 15 x + 9 x^{2} - x^{3} \Rightarrow

evaluate f(2):

f (2) = 7 - 15 \cdot 2 + 9 \cdot 2^{2} - 2^{3}

f (2) = 7 - 30 + 36 - 8 = 5 \Rightarrow

tangent at point: (2,5)
The slope of the tangent line to graph of a function at a point is equal to first derivative of the function evaluated at that point:

f^{'} (x) = - 15 + 18 x - 3 x^{2} = - 3 (x^{2} - 6 x + 5)

f'(2)=-3(4-12+5)=9
Slope = m = 9, point

(x_{1}, y_{1}) = (2, 5)

, using point-slope form:

y - y_{1} = m (x - x_{1}) :

y - 5 = 9 (x - 2) \Rightarrow

equation of tangent in point-slope form

y = 9 x - 13 \Rightarrow

equation of tangent in slope-intercept form

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