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 24E

(a)

To determine

To find: The equation of the tangent line to the curve at the point.

Expert Solution

Answer to Problem 24E

The equation of the tangent line to the curve y=3x+6cosx at (π3,π+3) is y=(333)x+3+π3_.

Explanation of Solution

Given:

The equation of the curve is, y=3x+6cosx and the point is (π3,π+3).

Derivative rules:

(1) Constant Multiple Rule: ddx[cf(x)]=cddxf(x)

(2) Power Rule: ddx(xn)=nxn1

(3) Sum Rule: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

The derivative of y is dydx, which is obtained as follows,

dydx=ddx(y) =ddx(3x2x3)

Apply the sum rule (3)

ddx[3x+6cosx]=ddx[3x]+ddx[6cosx]

Apply the constant multiple rule(1).

ddx[3x+6cosx]=3ddx[x]+6ddx[cosx]

Apply the power rule (2)and simplify the expressions,

ddx[3x+6cosx]=3(1x11)+6(sinx)=36sinx

Therefore, the derivative of the function y=3x+6cosx is 36sinx_.

The slope of the tangent line at (π3,π+3) is,

m=dydx|x=π3 =36sinπ3           (Qsinπ3=32)=36(32)=333

Thus, the slope of the tangent line at (π3,π+3) is m=333_.

Substitute (π3,π+3) for (x1,y1) and 333 for m in equation (1),

y(π+3)=(333)(xπ3)yπ3=(333)x(333)π3yπ3=(333)xπ+3πy3=(333)x+3π

Add 3 on both sides and simplify further,

y3+3=(333)x+3π+3y=(333)x+3π+3

Therefore, the equation of the tangent line to the curve y=3x+6cosx is y=(333)x+3+3π.

(b)

To determine

To sketch: The given curve and the tangent line at the given point (π3,π+3).

Expert Solution

Explanation of Solution

Given:

The curve is y=3x+6cosx and the tangent line at (π3,π+3) is y=(333)x+3+3π.

Graph:

Use the online graphing calculator to draw the graph of the functions as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.3, Problem 24E

From Figure 1, it is observed that the equation of the tangent line touches the curve y=3x+6cosx at the point (π3,π+3).

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