Q. 17 - Calculus

Question

Consider again the function f graphed at the left. At which values of x does f have the greatest instantaneous rate of change? The least? At which values of x is the
instantaneous rate of change of f equal to zero?

Step-by-Step Solution

Answer

The function has greatest instantaneous rate of change at $x=0.5$ and the least at $x=-1,2$ and constant at $x=-0.4,1.3$

Step 1 . Given Information.

The graph of the function.

Step 2. Find the greatest instantaneous rate of change.

The greatest instantaneous rate of change of the function is highest slope of the tangent to the given graph.

From the graph, the tangent at $x=0.5$ has the maximum slope.

The function has greatest instantaneous rate of change at $x=0.5$

Step 3. Find the least instantaneous rate of change.

The least instantaneous rate of change of the function is least slope of the tangent to the given graph.

From the graph, the tangent at $x=-1,2$ has the minimum slope.

The function has least instantaneous rate of change at $x=-1,2$

Step 4. Find the constant instantaneous rate of change.

When the slope is constant, the instantaneous rate of change is zero.

The function has zero instantaneous rate of change at $x=-0.4,1.3$