- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 3
Exponential and trigonometric functions practice question 20
Find every in the interval at which the graph of the function below has a horizontal tangent line, and enter the sum of those values.
Horizontal tangents are where . Which rule differentiates this product, and can you write in terms of alone?
. Replace by and factor the quadratic in .
Solve each factor for in , and do not cancel a factor that can be zero.
An exact expression, or a decimal to two places.
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Differentiate with the product rule, write the derivative in terms of sin x alone, and factor. Each factor gives its own solutions; keep them all.
Working
The product rule: (cos x)′ = −sin x and (1 + sin x)′ = cos x.
Use cos² x = 1 − sin² x, then factor 2sin² x + sin x − 1.
Every solution in (0, 2π).
At 3π/2 the derivative is zero but does not change sign: a horizontal tangent at an inflection point, which still counts.
Answer
The answer is five pi over 2, about 7.85.
Check
f′(π/6) = −(1/2)(3/2) + 3/4 = 0 and f′(3π/2) = −(−1)(0) + 0 = 0, while just either side of 3π/2 f′ is positive, so the graph flattens there without turning.
Where this goes wrong
Cancelling the factor sin x + 1, or never finding it, and so missing x = 3π/2. There sin x = −1 and f′ = 0, even though the graph does not turn.
Leans on
The product rule, from Differentiation rules
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