Let 𝐫(t)=lnt 𝐢+2 t 𝐣+t^2 𝐤 . Find (a) 𝐫^'(t) (b) (d s)/(d t) (c) ∫1^3𝐫^'(t) d t

Name: Problem 43, Chapter 12: Calculus Early Transcendentals
Uploaded: 2020-11-09T15:56:57-08:00
Duration: 1 min 34 s
Channel: Sriram Soundarrajan
Description: Let 𝐫(t)=lnt 𝐢+2 t 𝐣+t^2 𝐤 . Find (a) 𝐫^'(t) (b) (d s)/(d t) (c) ∫1^3𝐫^'(t) d t

step 1 Today we are going to solve problem number 43 here r of t equals l n t i plus 2 t j plus t squar

Let 𝐫(t)=lnt 𝐢+2 t 𝐣+t^2 𝐤 . Find (a) 𝐫^'(t) (b) (d...

Let $\mathbf{r}(t)=\ln t \mathbf{i}+2 t \mathbf{j}+t^{2} \mathbf{k} .$ Find
$$
\begin{array}{llll}{\text { (a) }\left\|\mathbf{r}^{\prime}(t)\right\|} & {\text { (b) } \frac{d s}{d t}} & {\text { (c) } \int_{1}^{3}\left\|\mathbf{r}^{\prime}(t)\right\| d t} & {}\end{array}
$$

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Key Concepts

Integration of the Speed Function

Integrating the speed function (the magnitude of the derivative) over a specified interval yields the total distance or arc length of the path described by the vector function. This approach is fundamental in problems involving the measurement of curves in space.

Arc Length of a Curve

Arc length is determined by integrating the norm of the derivative (speed) of a vector function over a given interval. This integral gives the cumulative distance traveled along the curve, linking the concepts of differentiation and integration in the context of vector calculus.

Vector Function Derivative

This concept involves differentiating each component of a vector-valued function with respect to its parameter. The derivative represents the velocity vector of a moving point along a curve, capturing its instantaneous rate of change in each coordinate direction.

Magnitude of a Vector

The magnitude (or norm) of a vector is a measure of its length, computed as the square root of the sum of the squares of its components. It is essential for determining how fast the function's value is changing, which is especially relevant when the vector represents velocity.

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