   Chapter 11, Problem 49RE 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 47-52, find the equation of the tangent line to the graph of the given equation at the specified point. y = x 2 e − x ; x = − 1

To determine

To calculate: The equation of the tangent to the graph of the equation y=x2ex at x=1.

Explanation

Given Information:

The provided equation is y=x2ex and x=1.

Formula used:

Slope of tangent to the graph of f(x) at point (x1,y1) is given by dydx|(x1,y1).

Product rule of derivative of differentiable functions, f(x) and g(x) is

ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x).

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

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

Equation of line is y=mx+b where m is the slope and b=y1mx1 when line passes through (x1,y1).

Calculation:

Consider the equation, y=x2ex

Find the slope of tangent line to the equation y=x2ex by determining the derivative.

Then, take ddx of both sides of the equation,

ddx(y)=ddx(x2ex)dydx=ddx(x2ex)

Apply the product rule of derivative,

dydx=ddx(x2)ex+(x2)ddx(ex)

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

Now, apply the above formula and take the derivative of (x2) which is equal to (2x),

dydx=2xex+(x2)exddx(x)=2xex+x2exddx(x)

Now, apply the constant multiple rule,

dydx=2xex+x2

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