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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable. Problems 41 and 42 explore this relationship. Use each given function and the indicated point to complete the following.

(a) Write the equation of the tangent line to the curve at the indicated point.

(b) Use a graphing calculator to graph both the function and its tangent line. Be sure your graph shows the point of tangency.

(c) Repeatedly zoom in on the point of tangency. Do the function and the tangent line eventually become indistinguishable?

f ( x ) = 3 x 2 + 2 x   at  x  = 1

(a)

To determine

To calculate: The equation of the tangent to curve f(x)=3x2+2x at x=1.

Explanation

Given Information:

The curve is provided by the function f(x)=3x2+2x, and the point is x=1.

Formula Used:

According to sum rule of derivatives,

If f(x)=u(x)+v(x)

Then, f(x)=u(x)+v(x)

According to power rule,

If f(x)=xn, then f(x)=nxn1.

Equation of tangent at any point (x1,y1) is given by,

yy1=y(xx1)

Calculation:

Consider the provided function,

f(x)=3x2+2x

Since, the slope of the tangent at any point of the curve is given by derivative of the function.

Therefore, to calculate slope of the tangent calculate the derivative of the function.

Apply the sum rule of derivatives,

y=ddx(3x2)+ddx(2x)

Use the power rule to find the derivative of the function,

y=ddx(3x2)+d

(b)

To determine

To graph: The function and the equation of the tangent to curve f(x)=3x2+2x at x=1

(c)

To determine

If both the curves become indistinguishable on repeatedly zooming for curve f(x)=3x2+2x at x=1 and the tangent y=8x3.

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