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What is the relationship between the derivative of a function fat a point x=c, the slope of the tangent line to the graph of fat x=c, and the instantaneous rate of change of fat x=c?

Short Answer

Expert verified

The derivative of fat x=c, the slope of the tangent line at x=cand instantaneous rate of change at x=call represent the same value i.e.f(c+h)-f(c)h.

Step by step solution

01

Step 1. Given information

A functionf(x).

02

Step 2. Explanation

The derivative of a function at any two instantaneous points x=cand x=c+his given by f(c+h)-f(c)h.

While the slope of the tangent line at x=cis also the rate of change of the function ffor two instantaneous points x=cand x=c+h.

Therefore, the slope of the tangent line is f(c+h)-f(c)h.

Thus the derivative of fat x=c, the slope of the tangent line at x=cand instantaneous rate of change atx=call represent the same value.

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