Question

Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table.

f (x) = x^5,       (2, 32)

Answer 1

Step 1 :

The function is and the point is .

Differentiate the function with respect to .

Power rule of derivatives : .

At the point , .

This is the slope of the tangent line.

Slope of the tangent line is .

Step 2 :

Point-slope form of the line equation is .

Substitute and in the above equation.

The tangent line equation is .

Step 3:

Use the linear approximation to complete the table.

	1.9	1.99	2	2.01	2.1
	24.741	31.208	32	32.808	40.841
	24.000	31.200	32	32.800	40.000

The table compares the values of y given by linear approximation with the values of near .

Comments

The graph of the tangent line is :

Observe the graph, the closer x is to 2 is the better the approximation.

The linear approximation of depends on the point of tangency.

At the different point on the graph of , we will obtain a different tangent line approximation.

Solution :

The tangent line is .

The table is :

	1.9	1.99	2	2.01	2.1
	24.741	31.208	32	32.808	40.841
	24.000	31.200	32	32.800	40.000