Finding r'(t) for r(t) = <t², sin(2t), e^t>

Calculus 3 · Axiom Academy

Differentiate a vector-valued function component-by-component, then use the result to find velocity, speed, and the unit tangent vector. Given the vector-valued function , find , evaluate it at t=0 , and use the result to find the unit tangent vector . Excellent work — you differentiated a vector-valued function and used it to analyze motion. Here's what to keep: Component-by-Component: — differentiate a vector-valued function by differentiating each component separately. Physical Interpretation: is the velocity vector — it points in the direction of motion, and its magnitude is the speed. Unit Tangent Vector: always has magnitude 1 — it gives direction only, with no speed information. Finding Extremes: a bounded, periodic component like moves fastest in its positive direction exactly when the trig factor reaches its own maximum, — which recurs every . Speed vs. Velocity: speed is (a scalar), while velocity is (a vector with direction). This foundation is essential for analyzing motion in three dimensions, finding arc length, and studying curvature — ideas that carry straight into physics, engineering, and the rest of vector calculus.

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