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Calculus/Curve sketching with derivatives.

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Expert: - 5/27/2008


Question
Let f(x) be the function defined by f(x)=sin^(2)x-sinx for 0 <(less than or equal to) x <(less than or equal to) 3pie (3.14).  Determine the exact values of the x-intercepts of the graph of f(x).  Use calculus to determine the intervals on which f(x) is increasing.  Find the x-values for which the tangent to f(x) is parallel to the horizontal axis.

Answer

Sine curve
Questioner:   harman
Category:  Calculus
Private:  No
 
Subject:  ap calculus
Question:  Let f(x) be the function defined by f(x)=sin^(2)x-sinx for 0 <(less than or equal to) x <(less than or equal to) 3pie (3.14).  Determine the exact values of the x-intercepts of the graph of f(x).  Use calculus to determine the intervals on which f(x) is increasing.  Find the x-values for which the tangent to f(x) is parallel to the horizontal axis.
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Hi, Harman,

Since you  didn't indicate what, if anything, you already did on these, I will assume you just don't know how to get started.

f(x) = sin^2(x) - sinx for 0 <= x <= 3pi   [really 3pi?]

Determine the exact values of the x-intercepts of the graph of f(x).  

Solve the equation    sin^2(x) - sinx = 0
Factor:  sin x(sin x - 1) = 0
Solve:  sin x = 0 and sin x = 1.
Now use your knowledge of the graph of sin x to get your values ->  x = 0, pi/2, pi, 2pi.

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Use calculus to determine the intervals on which f(x) is increasing.  
Find the x-values for which the tangent to f(x) is parallel to the horizontal axis.

Get f'(x) = 2 sin x cos x - cos x.

Set that = 0:

2 sin x cos x - cos x = 0

Factor (like the other)

cos x(2 sin x - 1) = 0

As before, use your knowledge of the graph of sin x (and cos x) to get your values.

Solve  cos x = 0,  -->  x = pi/2, 3pi/2

sin x = 1/2.  --> x = pi/6, 5pi/6


Finally, use those to SEPARATE regions where the graph is rising (increasing) from where the graph is falling.

(see the attached graph)

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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