(a)

The function is  and solution point is .

Graph.

Graph the function .

Plot the point .

Observe the graph.

Consider a point on the curve such that the graph appears to be linear.

One such point is .

Find the secant line equation.

The two points are and .

Slope of the secant line is .

Slope of the secant line is .

Point-Slope form of line equation: .

Substitute and in point-slope form.

Secant line equation is .

(b)

Equation of the tangent line is .

Consider .

Apply derivative on each side with respect to .

Substitute  in .

The point is which means that .

Tangent line equation is .

Substitute and in the tangent line equation.

Tangent line equation is .

Secant line equation is .

The secant line and tangent line appears to be same when the two points come closer.

Hence, the slope of the secant line approaches to tangent line at , as points come closer to .

(c)

Graph the function .

Graph the tangent line .

The tangent line  is the most accurate tangency point.

If the point of the tangency is moved, the approximation will become less accurate.

(d)

Complete the table.

The function is .

The tangent line equation is .

Observe the table.

We can conclude that as the point moves away, the accuracy of the approximation becomes less.

(a)

The graph is

Approximating point is .

Secant line equation is .

(b)

The slope of the secant line approaches to tangent line at , as points come closer to .

(c)

Graph of the function and tangent line is

If the point of the tangency is moved, the approximation will become less accurate.

(d)

As the point moves away, the accuracy of the approximation becomes less.