Power functions and first sketches, question 4 | Longhand

Topic: Power functions and first sketches
Difficulty: Introductory
Suggested time: 3 min
Marks: 2
Theory: UBC Math 100 §1.5Key results

Let $f(x) = x^{3} + 4\sin x$ , with $x$ in radians. Which term dominates close to $x = 0$ , and which dominates for large $|x|$ ?

The x³ term dominates in both places: sin x is never bigger than 1, so the 4 sin x term hardly matters anywhere.
The 4 sin x term dominates in both places, because its coefficient 4 is bigger than the 1 in front of x³.
Near 0 the 4 sin x term dominates, so the graph looks like the line y = 4x. For large |x| the x³ term dominates, and the graph stays within 4 of y = x³.
Near 0 the x³ term dominates, so the graph is flat at the origin. For large |x| the 4 sin x term dominates, so the graph keeps oscillating.

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Theory UBC Math 100 §1.5Key results

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