Math Problem Statement

Numerically estimate the slope of the line tangent to the graph of the function f at the given input value. Show the numerical estimation table with four estimates. (Round your answers for f(x) to six decimal places. Round your answers for the slope of the secant to four decimal places. Estimate f '(1) to the nearest hundredth.) f(x) = 3

x ; x = 1 x → 1− f(x) Slope of secant = f(1) − f(x) 1 − x x → 1+ f(x) Slope of secant = f(1) − f(x) 1 − x 0.9 1.1 0.99 1.01 0.999 1.001 0.9999 1.0001 f '(1)

Solution

We are asked to numerically estimate the slope of the line tangent to the function f(x)=3xf(x) = 3x at x=1x = 1. The slope of the tangent line at x=1x = 1 can be estimated by using

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Tangent Line
Numerical Estimation

Formulas

Slope of Secant: (f(1) − f(x)) / (1 − x)
Derivative approximation using secants

Theorems

Derivative as the limit of secant slopes

Suitable Grade Level

Grades 11-12 (High School Calculus)

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