   Chapter 11.5, Problem 67E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 67-72, find the equation of the straight line described. Use graphing technology to check your answers by plotting the given curve together with the tangent line.Tangent to y = e x log 2 x at the point ( 1 , 0 )

To determine

To calculate: The equation of the tangent to the graph of the function y=exlog2x at the point (1,0). Also, sketch the curve along with the tangent line.

Explanation

Given Information:

The provided function is y=exlog2x and the point is (1,0).

Formula used:

The derivative of e raised to a function is ddxeu=eududx.

The derivative of the log to base b of x is ddxlogbx=1xlnb.

Constant multiple rule of derivative of function f(x) is f'(cx)=cf'(x) where c is constant.

Calculation:

Consider the function, y=exlog2x

Find the slope of tangent line to the function xyy2=x23 by determining the derivative.

Then, the derivative of the functions is,

dydx=ddx(exlog2x)

Apply the product rule of derivative,

dydx=ddxexlog2x+exddx(log2x)

The derivative of the log to base b of x is ddxlogbx=1xlnb.

The derivative of e raised to x is ddxex=ex.

Now, apply the above formula and simplify the derivative,

dydx=exlog2x+ex1xln2=exlog2x+exln2

So, the slope of the tangent to the graph of function y=exlog2x is exlog2x+exln2.

Now, the slope at the point (x1,y1) is evaluated by calculating the value of dydx|(x1,y1).

Here, x1=1 and y1=0

Then, the slope at the point (1,0) is,

dydx|(1,0)=e0log2(0)+e1ln2=10+eln2=eln2

Thus, the slope of function y=exlog2x at the point (1,0) is eln2

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