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6-87.
  1. Consider the function f(x) = −3x . Homework Help ✎

    1. Find the equation of the tangent line to f(x) = −3x at x = 0.

    2. Approximate x = 0.1 using part (a).

    3. Use f ″(x) to justify whether part (b) is an under or over approximation of the actual value.

Evaluate f '(0) to find the slope, and evaluate f(0) to find the y-intercept. Use the point-slope formula to create the equation of the tangent line.

Use the tangent line at x = 0 to evaluate x = 0.1. This will approximate f(0.1). Why approximate?
Since a tangent is a linear equation, it is easier to evaluate than f(x).

A concave up graph has tangent lines that are below the curve. Sketch it, you will see.
And a concave down graph has tangent lines that are above the curve.

Overestimate since f ''(0) = −(ln 3)2 · 30 = −(ln 3)2 < 0