`f(x) = kx^4, y = 4x - 1` Find k such that the line is tangent to the graph of the function.

Textbook Question

Chapter 2, 2.2 - Problem 68 - 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 one point of the original function.  This means that:

`kx^4 = 4x-1` 

We have two variables that we don't know.

Find the derivative of `f(x)` .  The k is a constant and we can use the power rule.

`f'(x) = 4kx^3`

The slope of the tangent line is four, so we will plug that into this equation.

`4 = 4kx^3`

Divide four k on both sides.

`1/x^3=k`

Substitute k back into the first equation, .

`(1/x^3)x^4 = 4x-1`

`x = 4x-1`

`-3x=-1`

`x= 1/3`

Plug this back to to determine k.

`k(1/3)^4 = 4(1/3)-1`

`k(1/81)= 4/3-1`

`k(1/81)=1/3`

Multiply 81 on both sides.

`k= 81/3 = 27` 

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