Find the derivative of a vector-valued function representing a helix, evaluate it at a specific point, and describe how to sketch the resulting tangent vector.

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Derivative and Tangent Vector of a Helix

Multivariable calculus warmup

Let \(\vec{r}(t) = \langle \cos t, \sin t, t \rangle\). Compute \(\vec{r}'(t)\) and \(\vec{r}'(\pi/2)\). Sketch \(\vec{r}'(\pi/2)\) with its initial point at the origin and at \(\vec{r}(\pi/2)\).

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What you need first

Derivatives of sine and cosine
Evaluating trigonometric functions at pi/2
Understanding that vectors have magnitude and direction but no fixed position

What this problem tests

Tests the ability to differentiate basic trigonometric and polynomial components of a vector function, evaluate them, and understand the geometric interpretation of the derivative as a tangent vector.

This is a standard introductory problem in multivariable calculus or vector calculus, introducing the concept of differentiating vector functions and visualizing tangent vectors.

Derivative and Tangent Vector of a Helix

What you need first

What this problem tests

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