Find the equation of the tangent line T to the graph of f at the given point.

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)

asked Jan 24, 2015 in CALCULUS

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 31.2 32 32.8 40

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

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 31.2 32 32.8 40