The Derivative

Topics

Problems 2

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Topics Problems 2

Problem : Calculate the derivative of f (x) = x² at x = 1.

Substituting 1 for x₀ in the formula for the derivative, we have

f'(1)	=
	=
	=	2 + Δx
	=	2

Observing the graph of f (x) = x², we see that this is the slope of the tangent to the graph at the point (1, f (1)) = (1, 1).

Problem : Find the vertex of the parabola f (x) = x² + 2x + 2 using the derivative.

At the vertex, the tangent line to the graph will be horizontal, with slope 0. Therefore, we search for an x such that f'(x) = 0. We have

f'(x)	=	limΔx→0
	=	limΔx→0
	=	limΔx→0
	=	limΔx→02x + 2 + Δx
	=	2(x + 1)

Thus f'(x) = 0 if and only if x = - 1. Now f (x) = (- 1)² + 2(- 1) + 2 = 1, so the vertex of the parabola is (- 1, 1). We can check this by noting that f (x) - 1 = (x + 1)², so the graph of f (x) is the graph of x² translated 1 unit to the left and 1 unit up.

Problem : Find the equation of the tangent line to the graph of f (x) = x³ at x = 2.

First we compute f'(2):