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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

1-40 Use the guidelines of this section to sketch the curve.

y = sin x + 3 cos x ,     2 π x 2 π

To determine

To sketch:

The curve of y

Explanation

1) Concept:

i) A domain is the set of x values that satisfy the function.

ii) To find x-intercept, put y=0, and to find y-intercept, put x=0 in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if f-x=fx, then it is an even function, so it has y-axis symmetry. If f-x=-fx, then it is an odd function, so it has x-axis symmetry. And if f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

vi) The number f(c) is a local maximum value of f  if fcf(x) when x is near c and is a local minimum value of f if fc f(x) when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points

2) Given:

y=sinx+3cosx, -2π<x<2π

3) Calculations:

Here, first find the domain of the given function and the x & y intercepts. Next, check the symmetry, asymptotes, intervals of increase and decrease, local maximum and minimum values, concavity, and points of inflection. Using these, sketch the curve

A. Domain

fx is defined for all the values of the given interval

So Domain is -2π<x<2π

B. Intercepts:

For y intercept plug x=0  in the given function and solve it for y.

y=sin0+3cos0=3

The y-intercept is 0, 3

For x intercept, plug y=0 in the original function and solve it for x.

0=sinx+3cosx

sinx=-3cosx

tanx=-3

Therefore, x-intercepts are -4π3, 0, -π3, 0, 2π3, 0, 5π3, 0

C. Symmetry

For f-x replace x by (-x)

 f-x=sin-x+3 cos(-x)

f-x= -sinx+3 cosx

So f(x) has no symmetry

D. Asymptote

Here, the given function is bound between the given interval, so there are no horizontal and vertical asymptotes

E. Intervals of increase or decrease

To find the intervals of increase or decrease, find the derivative of the given function

f'x=cosx-3sinx

Now find where f'=0 or undefined

0=cosx-3sinx

cosx=3sinx

Divide both sides by sinx,

cotx=3

x=cot-1(3)

This is true for x=π6, 7π6, -5π6, -11π6(-2π, 2π)

 f(x) is decreasing at -11π6, -5π6, π6, 7π6

f(x) is increasing at -2π, -11π6, -5π6, π6, 7π6, 2π

F

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