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

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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 by anonymous

1 Answer

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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 image.

answered Jan 24, 2015 by lilly Expert

The graph of the tangent line is :

image

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

 

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