`y = x^4 - 3x^2 + 2, (1,0)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point,...

`y = x^4 - 3x^2 + 2, (1,0)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.

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Textbook Question

Chapter 2, 2.2 - Problem 53 - Calculus of a Single Variable (10th Edition, Ron Larson).
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kalau | (Level 2) Adjunct Educator

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Given:  

Take the derivative of this function.

`y' = 4x^3-6x`

Substitute the value of x=1 from the given point into the derivative function.

`y' = 4(1)^3-6(1)`

`y'=4-6 = -2`

The slope at the given point is negative 2.

a)

`y'=-2`

c)

Graph the derivative function :

The derivative at x=1 is negative 2.

b) Our slope is -2, and we will use the slope-intercept form with the given point to find the y-intercept, and write our equation.

`y=mx+b`

Substitute the point and slope.

`0 = (-2)(1)+b`

`0=-2+b`

`b=2`

The equation of the tangent line is then:

`y=-2x+2`

The original function and the tangent line are graphed in the imaged attached.

 

 

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