BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.3, Problem 22E
To determine

Find the equation of the tangent line usinggiven data.

Expert Solution

Answer to Problem 22E

The equation of the tangent line is y=x+1 .

Explanation of Solution

Given:

The given curve is y=1sinx+cosx and the given pointis (0,1) .

Calculation:

Find the derivative of the function.

  y=1sinx+cosx

Use quotient rule.

Use quotient rule.

  (fg)'=gf'fg'g2

  dydx=(sinx+cosx)ddx(1)1ddx(sinx+cosx)(sinx+cosx)2

Apply sum rule.

  (f+g)'=f'+g'

  dydx=(sinx+cosx)ddx(1)ddx(sinx)ddx(cosx)(sinx+cosx)2

Use derivative rule ddx(constant)=0 , ddx(sinx)=cosx and ddx(cosx)=sinx .

  dydx=(sinx+cosx)0cosx(sinx)(sinx+cosx)2dydx=sinxcosx(sinx+cosx)2

Substitute the value of x=0 in the derivative.

  dydx=sinxcosx(sinx+cosx)2=sin(0)cos(0)[sin(0)+cos(0)]2=01[0+1]2=1

Slope =1

Use point-slope form for the equation of tangent line.

  yy1=m(xx1)y1=1,x1=0,m=1y1=1(x0)y1=xy1+1=x+1y=x+1

Hence theequation of the tangent line is y=x+1 .

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