Exponential and trigonometric functions, question 18

Topic: Exponential and trigonometric functions
Difficulty: Challenging
Suggested time: 6 min
Marks: 4
Theory: UBC Math 100 §4.2 CLP-1 §2.8Key results

Find the constants $a$ and $b$ for which the function below is differentiable everywhere, and enter $a + b$ .

f(x) = \begin{cases} ax + b, & x < \dfrac{\pi}{4} \\[8pt] 3\sin x + 4\cos x, & x \geq \dfrac{\pi}{4} \end{cases}

Your answer

An exact value, or a decimal to two decimal places.

How sure are you?

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

Worked solution

Strategy

Both pieces are differentiable on their own; only the join at π/4 can fail. Matching slopes there fixes a; continuity then fixes b, remembering that the line’s value at the join is a(π/4) + b.

Working

$\text{slope at } \tfrac{\pi}{4}:\quad a = 3\cos\tfrac{\pi}{4} - 4\sin\tfrac{\pi}{4} = \tfrac{3\sqrt{2}}{2} - \tfrac{4\sqrt{2}}{2} = -\tfrac{\sqrt{2}}{2}$
The derivative of the right piece is 3cos x − 4sin x.
$\text{value at } \tfrac{\pi}{4}:\quad a\cdot\tfrac{\pi}{4} + b = 3\sin\tfrac{\pi}{4} + 4\cos\tfrac{\pi}{4} = \tfrac{7\sqrt{2}}{2}$
Continuity at the join.
$b = \tfrac{7\sqrt{2}}{2} - \left(-\tfrac{\sqrt{2}}{2}\right)\tfrac{\pi}{4} = \tfrac{7\sqrt{2}}{2} + \tfrac{\sqrt{2}\,\pi}{8}$
Solve with a = −√2/2.
$a + b = -\tfrac{\sqrt{2}}{2} + \tfrac{7\sqrt{2}}{2} + \tfrac{\sqrt{2}\,\pi}{8} = 3\sqrt{2} + \tfrac{\sqrt{2}\,\pi}{8} \approx 4.80$
Combine.

Answer

a = -\frac{\sqrt{2}}{2},\ b = \frac{7\sqrt{2}}{2} + \frac{\sqrt{2}\,\pi}{8},\ a + b = 3\sqrt{2} + \frac{\sqrt{2}\,\pi}{8}

Check

At x = 1 the line ax + b has height a + b, and as a tangent to a concave curve it sits just above 3 sin 1 + 4 cos 1 ≈ 4.69.

Where this goes wrong

Setting b equal to the value at the join and getting a + b = 3√2. The line is ax + b, so its value at π/4 is a(π/4) + b, and the a(π/4) part has to be moved across.

Leans on

Differentiability at a join, from The definition of the derivative

Try a similar question

Find the values of the constants

a

and

b

for which the function below is differentiable everywhere, and enter

b

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Exponential and trigonometric functions practice question 18