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Chapter 11: Problem 8

How do you determine whether \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is continuous at \(t=a ?\)

Short Answer

Expert verified
To determine the continuity of a vector function \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) at \(t=a\), you need to check the continuity of its component functions \(f(t)\), \(g(t)\), and \(h(t)\) at \(t=a\). If all component functions are continuous at \(t=a\), which means that \(\lim_{t\to a} f(t) = f(a)\), \(\lim_{t\to a} g(t) = g(a)\), and \(\lim_{t\to a} h(t) = h(a)\), then the vector function is continuous at \(t=a\).

Step by step solution

01

Check the continuity of the function f(t)

Determine if the limit of \(f(t)\) exists at \(t=a\) by calculating \(\lim_{t\to a} f(t)\). If this limit exists and is equal to \(f(a)\), that is, \(\lim_{t\to a} f(t) = f(a)\), then f(t) is continuous at \(t=a\).
02

Check the continuity of the function g(t)

Determine if the limit of \(g(t)\) exists at \(t=a\) by calculating \(\lim_{t\to a} g(t)\). If this limit exists and is equal to \(g(a)\), that is, \(\lim_{t\to a} g(t) = g(a)\), then g(t) is continuous at \(t=a\).
03

Check the continuity of the function h(t)

Determine if the limit of \(h(t)\) exists at \(t=a\) by calculating \(\lim_{t\to a} h(t)\). If this limit exists and is equal to \(h(a)\), that is, \(\lim_{t\to a} h(t) = h(a)\), then h(t) is continuous at \(t=a\).
04

Determine the continuity of the vector function

If Steps 1, 2, and 3 are all true and all component functions (f(t), g(t), and h(t)) are continuous at \(t=a\), then the vector function \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is continuous at \(t=a\).

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Most popular questions from this chapter

Let $$\mathbf{u}(t)=2 t^{3} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}-8 \mathbf{k} \text { and } \mathbf{v}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-e^{2 t} \mathbf{k}$$ Compute the derivative of the following functions. $$\mathbf{u}(t) \times \mathbf{v}(t)$$

Use the formula in Exercise 79 to find the (least) distance between the given point \(Q\) and line \(\mathbf{r}\). $$Q(-5,2,9) ; \mathbf{r}(t)=\langle 5 t+7,2-t, 12 t+4\rangle$$

Use vectors to show that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) is the point \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) (Hint: Let \(O\) be the origin and let \(M\) be the midpoint of \(P Q\). Draw a picture and show that $$\left.\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P}) \cdot\right)$$

Evaluate the following definite integrals. $$\int_{1 / 2}^{1}\left(\frac{3}{1+2 t} \mathbf{i}-\pi \csc ^{2}\left(\frac{\pi}{2} t\right) \mathbf{k}\right) d t$$

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle t^{4}-3 t, 2 t-1,10\right\rangle$$
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