Exponential and trigonometric functions, question 21

Topic: Exponential and trigonometric functions
Difficulty: Capstone
Suggested time: 10 min
Marks: 5
Theory: UBC Math 100 §4.2 CLP-1 §2.8Key results

The function below is differentiable everywhere. Find $a$ .

f(x) = \begin{cases} ax + b, & x < \dfrac{\pi}{6} \\[8pt] \dfrac{5\cos x}{3 + \sin x + \cos x}, & x \geq \dfrac{\pi}{6} \end{cases}

Your answer

An exact value, or a decimal to three places.

How sure are you?

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Theory UBC Math 100 §4.2 CLP-1 §2.8Key results

Worked solution

Strategy

A line joined to a smooth function is differentiable at the join exactly when the line has the function’s value and slope there. Matching slopes makes a the derivative of the quotient at π/6; matching values would then fix b, which is not asked for.

Working

$u = 5\cos x,\ u' = -5\sin x;\qquad v = 3 + \sin x + \cos x,\ v' = \cos x - \sin x$
Parts for the quotient rule.
$g'(x) = \frac{\begin{gathered} -5\sin x\,(3 + \sin x + \cos x) \\ {} - 5\cos x\,(\cos x - \sin x) \end{gathered}}{(3 + \sin x + \cos x)^{2}}$
u′v − uv′ over v².
$= \frac{-15\sin x - 5\sin^{2}x - 5\cos^{2}x}{(3 + \sin x + \cos x)^{2}} = -\frac{15\sin x + 5}{(3 + \sin x + \cos x)^{2}}$
The sin x cos x terms cancel, and 5sin² x + 5cos² x = 5.
$g'\!\left(\tfrac{\pi}{6}\right) = -\frac{\tfrac{15}{2} + 5}{\left(\tfrac{7 + \sqrt{3}}{2}\right)^{2}} = -\frac{50}{52 + 14\sqrt{3}} = -\frac{25}{26 + 7\sqrt{3}}$
At π/6, sin x = 1/2 and cos x = √3/2, so v = (7 + √3)/2; and (7 + √3)² = 52 + 14√3.
$a = g'\!\left(\tfrac{\pi}{6}\right) = -\frac{25}{26 + 7\sqrt{3}} \approx -0.656$
Matching slopes gives a; matching values would give b, which is not asked for.

Answer

a = -\dfrac{25}{26 + 7\sqrt{3}} \approx -0.656

Check

The quotient goes from 1.25 at x = 0 to about 0.573 at x = π/3, an average slope of about −0.65, near −0.656 at the midpoint π/6.

Where this goes wrong

Differentiating only the numerator, which gives −5sin x/(3 + sin x + cos x) and a ≈ −0.573. The quotient rule’s second term, −u v′, is not zero at π/6 and has to be kept.

Leans on

Differentiability at a join, from The definition of the derivative

Try a similar question

Find the constants

a

and

b

for which the function below is differentiable everywhere, and enter

a + b

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Exponential and trigonometric functions practice question 21