Derivatives Exercises

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• f'(a) is zero because the tangent line to f at a is horizontal (has a slope of 0).

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• f'(a) is negative because the tangent lint to f at a slopes downwards:

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• f'(a) is negative because the tangent lint to f at a slopes downwards:

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• f'(a) is zero because the tangent line to f at a is horizontal (has a slope of 0).

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• f'(a) is positive because the tangent line to f at a slopes upwards:

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• Draw the tangent lines to f at x₁ and x₂:

The slope of the tangent line to f at x₁, also known as f ' (x₁), looks to be about 0. The slope of the tangent line to f at x₂, also known as f ' (x₂), looks decidedly more negative; f ' (x₁) is definitely greater.

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• Draw the tangent lines to f at x₁ and x₂:

The slope of the tangent line to f at x₁, also known as f ' (x₁), looks positive but shallow. The slope of the tangent line to f at x₂, also known as f'(x₂), looks positive and much steeper. It's safe to say f ' (x₂) > f ' (x₁).

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• Draw the tangent lines to f at x₁, x₂, x₃:

The slope of the tangent line to f at x₁, also known as f ' (x₁), is positive and fairly steep.The slope of the tangent line to f at x₂, also known as f ' (x₂), looks like it's about zero.The slope of the tangent line to f at x₃, also known as f ' (x₃), is negative. Thereforef ' (x₃) < f ' (x₂) < f ' (x₁).

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• We'll start by drawing the tangent lines to f at x₁, x₂, and 0:

The slope of the tangent line to f at x₁, also known as f ' (x₁), is positive and fairly steep.The slope of the tangent line to f at x₂, also known as f ' (x₂), is negative and fairly steep.The slope of the tangent line to f at 0, also known as f ' (0), is zero. Thereforef ' (x₂) < f ' (0) < f ' (x₁).