Find when a car moves left or right by analyzing the sign of its velocity function.

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Direction of Motion from a Cubic Position Function

Rectilinear and curvilinear motion warmup

An automobile moves along a straight highway, with its position \( s \) given by \( s = 12t^3 - 18t^2 + 9t - 1.5 \) (\( s \) in feet, \( t \) in seconds). When is the car moving to the right, when to the left, and where and when does it change direction?

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What you need first

Power rule for differentiation
Factoring quadratic expressions
Relationship between position, velocity, and direction of motion

What this problem tests

Tests the ability to compute a derivative, factor a quadratic, and interpret the sign of the derivative in a physical context.

This is a standard introductory calculus problem demonstrating the physical interpretation of the first derivative.

Direction of Motion from a Cubic Position Function

What you need first

What this problem tests

More on Rectilinear and curvilinear motion