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Chapter 12: Problem 2

Find the given limits. $$\lim _{t \rightarrow-1}\left[t^{3} \mathbf{i}+\left(\sin \frac{\pi}{2} t\right) \mathbf{j}+(\ln (t+2)) \mathbf{k}\right]$$

Short Answer

Expert verified
The limit is \(-\mathbf{i} - \mathbf{j}.\)

Step by step solution

01

Identify the Limit Components

The vector function is given as \( \lim _{t \rightarrow -1} \left[ t^{3} \mathbf{i} + \left( \sin \frac{\pi}{2} t \right) \mathbf{j} + (\ln (t+2)) \mathbf{k} \right] \). We need to evaluate each component separately as \( t \) approaches \( -1 \).
02

Calculate the Limit for \( t^3 \mathbf{i} \)

Substitute \( t = -1 \) into \( t^3 \mathbf{i} \): \[ \lim_{t \rightarrow -1} t^3 \mathbf{i} = (-1)^3 \mathbf{i} = -\mathbf{i}. \]
03

Calculate the Limit for \( \sin \frac{\pi}{2} t \mathbf{j} \)

Substitute \( t = -1 \) into \( \sin \left( \frac{\pi}{2} t \right) \mathbf{j} \): \[ \lim_{t \rightarrow -1} \sin \left( \frac{\pi}{2} t \right) \mathbf{j} = \sin \left( \frac{\pi}{2}(-1) \right) \mathbf{j} = \sin(-\frac{\pi}{2}) \mathbf{j} = -1 \mathbf{j} = -\mathbf{j}. \]
04

Calculate the Limit for \( \ln(t+2) \mathbf{k} \)

Substitute \( t = -1 \) into \( \ln(t+2) \mathbf{k} \): \[ \lim_{t \rightarrow -1} \ln(t+2) \mathbf{k} = \ln((-1)+2) \mathbf{k} = \ln(1) \mathbf{k} = 0 \mathbf{k}. \]
05

Combine the Results

Combine the results from each limit component to find the overall limit: \[ \lim _{t \rightarrow -1} \left[ t^{3} \mathbf{i} + \left( \sin \frac{\pi}{2} t \right) \mathbf{j} + (\ln (t+2)) \mathbf{k} \right] = -\mathbf{i} - \mathbf{j} + 0 \mathbf{k}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Functions
A vector function is a function that takes one or more variables, often time or a scalar, and produces a vector as its output. The output vector has multiple components, typically represented by i, j, k in three-dimensional space, corresponding to each dimension's direction. Each component is associated with a part of the vector.
  • Components: For example, in the vector function \( t^3 \mathbf{i} + \sin\left(\frac{\pi}{2} t\right) \mathbf{j} + \ln(t+2) \mathbf{k} \), the expression \( t^3 \mathbf{i} \) represents the component in the x-direction (i-component).
  • Notation: Each component is multiplied by their respective unit vectors, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), which denote directions along the x, y, and z axes respectively.
  • Behavior: By evaluating each component's behavior individually as the variable changes, we gain insights into the behavior of the entire vector function.
Understanding vector functions is crucial in fields like physics and engineering, where you model phenomena over time or space.
Limit Computation
The limit of a function at a point helps us understand its behavior as it approaches that point. To compute the limit of a vector function, you evaluate the limit of each component individually, then combine them.
  • Evaluating Components: For the given vector function \( t^3 \mathbf{i} + \left( \sin \frac{\pi}{2} t \right) \mathbf{j} + (\ln (t+2)) \mathbf{k} \), find the limit of each part separately as \( t \rightarrow -1 \).
  • Combining Results: After computing each component's limit, compile them to find the complete limit. In the example, we find \(-\mathbf{i} - \mathbf{j} \) as the final result.
  • Continuous Functions: Understand that if each component function is continuous at the point you're evaluating, the limit calculation becomes straightforward by direct substitution.
Limits are foundational in calculus and crucial for solving real-world problems involving changing conditions.
Trigonometric Limits
Trigonometric limits often involve functions like sine, cosine, and tangent. Understanding how these functions behave as their input approaches a given point is essential.
  • Periodic Nature: Sine and cosine functions oscillate between -1 and 1. When evaluating their limits, consider their periodic properties.
  • Example Calculation: For \( \sin \left( \frac{\pi}{2} t \right) \), you substitute \( t = -1 \), and calculate \( \sin(-\frac{\pi}{2}) \), which gives \(-1\).
  • Continuous Functions: Sine and cosine are continuous and well-behaved across their domain, which simplifies direct limit computations.
Trigonometric limits are vital in fields like engineering and physics, where periodic functions model a wide range of phenomena.

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Most popular questions from this chapter

Products of scalar and vector functions Suppose that the scalar function \(u(t)\) and the vector function \(\mathbf{r}(t)\) are both defined for \(a \leq t \leq b.\) a. Show that ur is continuous on \([a, b]\) if \(u\) and \(r\) are continuous on \([a, b].\) b. If \(u\) and \(r\) are both differentiable on \([a, b],\) show that \(u \mathbf{r}\) is differentiable on \([a, b]\) and that $$\frac{d}{d t}(u \mathbf{r})=u \frac{d \mathbf{r}}{d t}+\mathbf{r} \frac{d u}{d t}$$

Find the are length parameter along the curve from the point where \(t=0\) by evaluating the integral \(s(t)=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau\) from Equation (3). Then use the formula for \(s(t)\) to find the length of the indicated portion of the curve. $$r(t)=(4 \cos t) \mathbf{i}+(4 \sin t) \mathbf{j}+3 t \mathbf{k}, \quad 0 \leq t \leq \pi / 2$$

Use a CAS to perform the following steps. a. Plot the space curve traced out by the position vector \(\mathbf{r}\). b. Find the components of the velocity vector \(d \mathbf{r} / d t\) c. Evaluate \(d \mathbf{r} / d t\) at the given point \(t_{0}\) and determine the equation of the tangent line to the curve at \(\mathbf{r}\left(t_{0}\right)\) d. Plot the tangent line together with the curve over the given interval. $$\begin{aligned} &\mathbf{r}(t)=\left(\ln \left(t^{2}+2\right)\right) \mathbf{i}+\left(\tan ^{-1} 3 t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}\\\ &-3 \leq t \leq 5, \quad t_{0}=3 \end{aligned}$$

Show that if \(\mathbf{u}\) is a unit vector, then the are length parameter along the line \(\mathbf{r}(t)=P_{0}+t \mathbf{u}\) from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right),\) where \(t=0,\) is \(t\) itself.

Volleyball A volleyball is hit when it is 4 \(\mathrm{ft}\) above the ground and 12 ft from a \(6-f t\) -high net. It leaves the point of impact with an initial velocity of \(35 \mathrm{ft} / \mathrm{sec}\) at an angle of \(27^{\circ}\) and slips by the opposing team untouched. a. Find a vector equation for the path of the volleyball. b. How high does the volleyball go, and when does it reach maximum height? c. Find its range and flight time. d. When is the volleyball 7 ft above the ground? How far (ground distance) is the volleyball from where it will land? e. Suppose that the net is raised to 8 ft. Does this change things? Explain.
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