11.2.2What do $2^{\text{nd}}$ derivatives represent in component form?

The parametric function $x(t) = e^{2t}$, $y(t) = t^2 + 3$ represents the position of a particle moving in an $xy$-plane.

1. Express the velocity of the particle at any time, $t$, as a parametric function where $t ≥ 0$.

2. Express the velocity of the particle at any time $t$ as a vector-valued function.

3. Write an expression for speed of the particle, $\mathbf{\| \vec { v } ( t ) \|}$, at any time $t$.

4. Write a equation for the slope of the parametric curve, $\frac { d y } { d x }$, at any time $t$.

5. Does $\frac { d y } { d x }$ represent the velocity of the particle at any time $t$? Explain.

A team of Calculus students is admiring a small ball (made of chalk) as it rolls around a chalkboard that is laying flat on the ground. As it rolls, the path of the chalk can be modeled by the parametric function $x(t) = 99t^{1/3}$, $y(t) = 20t^{1/2}$ for any time $t ≥ 0$. Unfortunately, after about one minute, the chalk marks became very faint and hard to see. At $t = 64$ seconds, no chalk remains. Naturally, the students became curious about what happens at $t = 64$.

1. Roberta is curious about the acceleration of the ball at $t = 64$. Write an acceleration vector, $\vec { a } ( t )$, that corresponds with this parametric function. Then use it to evaluate $\vec { a } ( 64 )$. Use the results to describe the acceleration at $t = 64$. 

2. Eleanor is curious about the concavity of the curve at $t = 64$, so she attempts to find its second derivative, $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . Examine her work at right.
    
   Roberta objects, “What you found is $\frac { d ^ { 2 } y } { d ^ { 2 } x }$, but you were supposed to find $\frac { d ^ { 2 } y } { d x ^ { 2 }}$. I do not think that these are the same thing!” 

   $\large\left. \begin{array} { l } { \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { \frac { d ^ { 2 } y } { d t ^ { 2 } } } { d t ^ { 2 } } = \frac { \frac { - 5 } { 3 / 2 } } { \frac { - 22 } { t ^ { 2 / 3 } } } = \frac { 5 } { 22 } t ^ { 1 / 6 } } \\ { \frac { d ^ { 2 } y } { d x ^ { 2 } } | _ { t = 64 } = \frac { 5 } { 11 } } \end{array} \right.$

   With your team, compare and contrast the expression $\frac { d ^ { 2 } y } { d ^ { 2 } x }$ with $\frac { d ^ { 2 } y } { d x ^ { 2 } }$. Which did Eleanor find? Which is the appropriate way to analyze the concavity of the curve? Be prepared to explain your answer to the class.

3. Help Eleanor out. With your team, find a method to calculate the concavity of the graph of the marble’s path at $t = 64$. Is it concave up or concave down at this moment? Show all steps.

4. Does the value you found in part (c) represent the acceleration of the particle at $t = 64$? Explain.

The path of a particle moving in an $xy$-plane can be traced by the parametric function $x(t)$, $y(t)$ where $t ≥ 0$. Not much is known about this particle except that it was at the origin when $t = 0$ and its horizontal velocity can be determined by $x′(t) = 2 ^ { \sqrt { t } }$. Its vertical velocity is not given. Also, the slopes of the lines tangent to the curve can be calculated using the function $f(t) = t^2$.

1. Write an equation for $y′\left(t\right)$, the vertical velocity of the particle at any time $t$.

2. What is the acceleration vector when $t = 1$? 

3. What is the concavity of the curve when $t = 1$?

Pete Moss is reviewing his notes from his football coach, shown below. He knows he needs to catch the ball when $t = 6$, three seconds after the quarterback throws the pass. Homework Help ✎

1. Write the general velocity and acceleration vectors for the ball’s flight.

2. Calculate the velocity and acceleration of the ball when the catch is made.

Calculate the area inside both $r = 4\cos\left(θ\right)$ and $r = 4\sin\left(θ\right)$. 11-57 HW eTool (Desmos). Homework Help ✎

Thoroughly investigate the graph of $y = e^{x}\cos\left(x\right)$ over the interval $[–1, 5]$. Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection and intercepts and provide the graphs of $y = f ′\left(x\right)$ and $y = f^{\prime\prime}(x)$. Be sure to justify all statements both graphically and analytically. 11-58 HW eTool (Desmos). Homework Help ✎

For what values of $x$ does $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { n ! }$ converge? Homework Help ✎

Multiple Choice: If $f(x) = 3x^{1/3} - 4x$ and $x_{1} = 1$, then Newton’s Method of approximating roots will result in this approximate value for $x_{2}$. Homework Help ✎