   Chapter 9.4, Problem 42E ### Mathematical Applications for the ...

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

#### Solutions

Chapter
Section ### 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 ) = 4 x − x 2  at  x  = 5

(a)

To determine

To calculate: The equation of the tangent line to curve f(x)=4xx2 at x=5.

Explanation

Given Information:

The provided function is f(x)=4xx2 and the provided point is x=5.

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.

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

yy1=y(xx1)

Calculation:

The provided function is f(x)=4xx2.

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

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

Apply the sum rule of derivatives,

f(x)=4xx2f(x)=ddx(4xx2)y=ddx(4x)ddx(x2)

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

y=ddx(4x)ddx(x2)=(4(1)x11)+(2x21)=42x

To determine the slope of the tangent at x=5

(b)

To determine

To graph: The function f(x)=4xx2 and the tangent line y=6x+25.

(c)

To determine

If both the curves f(x)=4xx2 and y=6x+25 become indistinguishable at x=5.

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