Consider the function f(x) = −3x . Homework Help ✎
Find the equation of the tangent line to f(x) = −3x at x = 0.
Approximate x = 0.1 using part (a).
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