Question

which of the following is the best approximation of the slope of the tangent line to the graph of a function f(x) at x=5? the slope of the secant line to the graph of f(x) at x=4.3 and x=5 the slope of the secant line to the graph of f(x) at x=4.5 and x=5 the slope of the secant line to the graph of f(x) at x=4.7 and x=5 the slope of the secant line to the graph of f(x) at x=4.9 and x=5 the slope of the secant line to the graph of f(x) at x=4.1 and x=5

Explanation:

Step1: Recall tangent-secant approximation

The slope of the tangent line at $x=a$ is approximated by the slope of a secant line through $(a, f(a))$ and $(x, f(x))$, where $x$ is as close to $a$ as possible.

Step2: Compare distances to $x=5$

Calculate the absolute difference between each $x$-value and 5:

• $|4.3 - 5| = 0.7$
• $|4.5 - 5| = 0.5$
• $|4.7 - 5| = 0.3$
• $|4.9 - 5| = 0.1$
• $|4.1 - 5| = 0.9$

The smallest difference is 0.1, corresponding to $x=4.9$.

Answer:

the slope of the secant line to the graph of f(x) at x=4.9 and x=5