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Numerade Educator

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Problem 4 Medium Difficulty

(a) Find the slope of the tangent line to the curve $ y = x - x^3 $ at the point $ (1, 0) $
(i) using Definition 1 (ii) using Equation 2

(b) Find an equation of the tangent line in part (a).

(c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at $ (1, 0) $ until the curve and the line appear to coincide.

Answer

a) (i) -2
(ii) -2
b) $y=-2 x+2$
c) graph unavailable

Discussion

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NH

Nadia H.

September 21, 2020

What is a numerator and denominator?

CR

Cam R.

September 21, 2020

Well the top number in a fraction. Shows how many parts we have. (The bottom number is the Denominator and shows how many equal parts the item is divided into.)

ST

Samantha T.

September 21, 2020

Can someone explain what the tangent line is?

HC

Howie C.

September 21, 2020

Hey Samantha In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve.

LP

Lindsey P.

September 21, 2020

Anyone else confused by the linear function, can someone explain?

AG

Alex G.

September 21, 2020

I know this one! In mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one.

Video Transcript

So now we have to function Y equals x minus x acute graft. That is the blue curve over here and then here uh the green line is our a tangent line. So if we look at uh the green tangent line and the blue function, the blue curve you could see it is just touching, the green tangent line just touches uh your blue function, your x minus execute function right here at the 0.1 comma zero. And so uh this green line Y equals negative two, X plus two is tangent to the function Y equals x minus X cubed right here at the 20.1 comma zero. And if we zoom in you'll continue to see that it's just grazing it right there at that point.