`f(x) = k - x^2, y = -6x + 1` Find k such that the line is tangent to the graph of the function.

Textbook Question

Chapter 2, 2.2 - Problem 63 - Calculus of a Single Variable (10th Edition, Ron Larson).
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kalau | (Level 2) Adjunct Educator

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The tangent line will touch a point on the function f(x). 

Set both equations equal to each other.

`k-x^2= -6x+1`

With two unknown variables, we will need another relationship of x and k.

Take the derivative of f(x) and set the derivative function equal to the slope of the tangent line equation.  The slope of the tangent line is negative 6.

`f'(x)=0-2x`

`-6 = -2x`

`x=3`

We can find k by substituting back the value of x into the first equation.

`k-(3)^2= -6(3)+1`

`k-9 = -18+1`

`k=-18+1+9`

`k=-8`

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