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Chapter 14: Problem 53

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle t^{2}+1, t+1,1\right\rangle$$

Short Answer

Expert verified
Question: Find the first, second, and third derivatives of the vector function \(\mathbf{r}(t) = \left\langle t^2+1, t+1, 1 \right\rangle\). Answer: The first, second, and third derivatives of the vector function \(\mathbf{r}(t)\) are: 1. \(\mathbf{r}'(t) = \left\langle 2t, 1, 0 \right\rangle\) 2. \(\mathbf{r}''(t) = \left\langle 2, 0, 0 \right\rangle\) 3. \(\mathbf{r}'''(t) = \left\langle 0, 0, 0 \right\rangle\)

Step by step solution

01

Compute the first derivative of the vector function

To find the first derivative of the vector function, differentiate each component of the function with respect to \(t\). $$\mathbf{r}'(t) = \frac{d\mathbf{r}}{dt} = \left\langle \frac{d(t^2+1)}{dt}, \frac{d(t+1)}{dt}, \frac{d(1)}{dt}\right\rangle$$ Calculate each component: $$\frac{d(t^2+1)}{dt} = 2t$$ $$\frac{d(t+1)}{dt} = 1$$ $$\frac{d(1)}{dt} = 0$$ So, the first derivative is: $$\mathbf{r}'(t) = \left\langle 2t, 1, 0 \right\rangle$$
02

Compute the second derivative of the vector function

Now, we will compute the second derivative of the vector function by finding the derivative of each component of the first derivative with respect to \(t\). $$\mathbf{r}''(t) = \frac{d\mathbf{r}'}{dt} = \left\langle \frac{d(2t)}{dt}, \frac{d(1)}{dt}, \frac{d(0)}{dt}\right\rangle$$ Calculate each component: $$\frac{d(2t)}{dt} = 2$$ $$\frac{d(1)}{dt} = 0$$ $$\frac{d(0)}{dt} = 0$$ So, the second derivative is: $$\mathbf{r}''(t) = \left\langle 2, 0, 0 \right\rangle$$
03

Compute the third derivative of the vector function

Finally, we will compute the third derivative of the vector function by finding the derivative of each component of the second derivative with respect to \(t\). $$\mathbf{r}'''(t) = \frac{d\mathbf{r}''}{dt} = \left\langle \frac{d(2)}{dt},\frac{d(0)}{dt}, \frac{d(0)}{dt}\right\rangle$$ Calculate each component: $$\frac{d(2)}{dt} = 0$$ $$\frac{d(0)}{dt} = 0$$ $$\frac{d(0)}{dt} = 0$$ So, the third derivative is: $$\mathbf{r}'''(t) = \left\langle 0, 0, 0 \right\rangle$$ To summarize the results, we have: $$\mathbf{r}'(t) = \left\langle 2t, 1, 0 \right\rangle$$ $$\mathbf{r}''(t) = \left\langle 2, 0, 0 \right\rangle$$ $$\mathbf{r}'''(t) = \left\langle 0, 0, 0 \right\rangle$$

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Key Concepts

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

Derivatives of Vector-Valued Functions
Vector-valued functions are an essential part of vector calculus. They are functions that take a single parameter, usually denoted as \( t \), and return a vector. Each component of these vector functions is typically a function of \( t \). To understand how vector-valued functions change, we need to compute their derivatives, just like with scalar functions. This involves differentiating each vector component separately with respect to the parameter \( t \).
For example, if you have a vector-valued function \( \mathbf{r}(t) = \left\langle f(t), g(t), h(t) \right\rangle \), its derivative \( \mathbf{r}'(t) \) is obtained by differentiating each component:
  • \( \frac{df}{dt} \)
  • \( \frac{dg}{dt} \)
  • \( \frac{dh}{dt} \)
Combining these derivatives gives us \( \mathbf{r}'(t) = \left\langle \frac{df}{dt}, \frac{dg}{dt}, \frac{dh}{dt} \right\rangle \). Each of these operations is similar to the differentiation rules you鈥檙e used to, like the power rule or chain rule in calculus.
Vector Functions
Vector functions can describe many physical phenomena, such as the position of a moving object, the velocity of a fluid, or the force on a body. A vector function maps parameters (usually time) to vectors, which can be points in space or physical quantities.
A typical vector function might look something like \( \mathbf{r}(t) = (x(t), y(t), z(t)) \). Here, the vector \( \mathbf{r}(t) \) changes as \( t \) changes. Each component \( x(t) \), \( y(t) \), and \( z(t) \) modifies the vector鈥檚 path in some space.
When working with vector functions, it's crucial to consider each part of the vector:
  • \( x(t) \) moves along the x-axis
  • \( y(t) \) moves along the y-axis
  • \( z(t) \) moves along the z-axis
Together, these components track how something behaves over time or through space. This makes vector functions versatile in modeling real-domain problems.
Higher-Order Derivatives
When dealing with vector functions, understanding higher-order derivatives is key to analyzing the behavior and characteristics of the function over time. A first derivative gives you the rate of change or velocity; the second derivative provides the acceleration, crucial for understanding dynamics.
Continuing this differentiation process leads to higher derivatives:
  • First derivative (\( \mathbf{r}'(t) \)): Shows how quickly and in which direction the vector function is changing over time.
  • Second derivative (\( \mathbf{r}''(t) \)): Indicates how the rate of change itself is changing, i.e., accelerating.
  • Third derivative (\( \mathbf{r}'''(t) \)): Occasionally known as "jerk," represents the change in acceleration.
While working through higher derivatives may seem repetitive, each layer uncovers new insights into the function's dynamic behavior. When all higher-order derivatives equal zero, it suggests the function has stabilized and is not deviating any further over time.

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Most popular questions from this chapter

Three-dimensional motion Consider the motion of the following objects. Assume the \(x\) -axis points east, the \(y\) -axis points north, the positive z-axis is vertical and opposite g. the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for \(t \geq 0\). b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. A baseball is hit 3 ft above home plate with an initial velocity of \(\langle 30,30,80\rangle \mathrm{ft} / \mathrm{s} .\) The spin on the baseball produces a horizontal acceleration of the ball of \(5 \mathrm{ft} / \mathrm{s}^{2}\) in the northward direction.

Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a ficnction of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. $$\mathbf{r}(t)=\langle t, \sin t\rangle, \text { for } 0 \leq t \leq \pi \quad \text { (sine curve) }$$

Consider a particle that moves in a plane according to the function \(\mathbf{r}(t)=\left\langle\sin t^{2}, \cos t^{2}\right\rangle\) with an initial position (0,1) at \(t=0\) a. Describe the path of the particle, including the time required to return to the initial position. b. What is the length of the path in part (a)? c. Describe how the motion of this particle differs from the motion described by the equations \(x=\sin t\) and \(y=\cos t\) d. Consider the motion described by \(x=\sin t^{n}\) and \(y=\cos t^{n}\) where \(n\) is a positive integer. Describe the path of the particle, including the time required to return to the initial position. e. What is the length of the path in part (d) for any positive integer \(n ?\) f. If you were watching a race on a circular path between two runners, one moving according to \(x=\sin t\) and \(y=\cos t\) and one according to \(x=\sin t^{2}\) and \(y=\cos t^{2},\) who would win and when would one runner pass the other?

Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a ficnction of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. $$\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle, \text { for }-2 \leq t \leq 2 \quad \text { (parabola) }$$

Prove that the line \(\mathbf{r}(t)=\left\langle x_{0}+a t, y_{0}+b t, z_{0}+c t\right\rangle\) is parameterized by arc length, provided \(a^{2}+b^{2}+c^{2}=1\)
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