Rapid Recall
- Write the derivative of the following functions:
- $f(x) = \pi^2$
- $\displaystyle g(x) = \frac{22}{7} x $
- $p(x) = x^3-2x^2+x+5$
Answer:
- $\displaystyle \frac{df}{dx} = 0$
- $\displaystyle g'(x) = \frac{22}{7}$
- $p'(x) = 3x^2-4x+1$
Where can I go for practice if my derivative skills are lacking? Thomas's calculus section 3.5 (the chain rule section) is a great place to practice. If you need to start in section 3.1, perfect. You can use any calc text, if you had calculus at another school.
- If a rover's motion on flat ground is described by $(x,y)=(3,4)t+(-2,5)$, what is its speed? Assume $x$ and $y$ are measured in meters and $t$ is given in minutes. (Hint: the rover's speed is constant in this situation.)
Answer:
When $t=0$, $(x,y) = (-2,5)$. When $t=1$, it is at $(1,9)$, so its position has changed by $\Delta x = 3$ and $\Delta y=4$. This means the rover has moved a total of $\sqrt{3^2+4^2}=5$ meters, so its speed is 5 meters/minute.
- Using the vectors on the board, draw $\vec u+\vec v$ and $\vec u-\vec v$.
Answer:
We'll discuss on the board.
- Give the component form of a vector of length 2 that is parallel to $\left<3,4\right>$.
Answer:
A unit vector is $\frac{1}{5}(3,4)$, so the desired vector is twice this, namely $$\frac{2}{5}(3,4) = \left(\frac{6}{5},\frac{8}{5}\right).$$
- Draw and shade the region in space satisfying $0\leq z\leq 5$ with $x=2$.
Answer:
The region is an infinitely long sheet with $0\leq z\leq 5$, located 2 units out on $x$-axis, with the $y$ value taking on all possible values. The most common error here is to not let $y$ take on all possible values, in which case you'll end up with a short line segment at $y=0$ and $x=2$ running from 0 to 5 in the $z$ direction.
Group Problems
(Don't forget to PTC after each problem)
- For this one, have everyone in the group use chalk at the same time. In 3D, plot the points $(1,2,0)$, $(1,0,3)$, $(0,2,3)$, $(1,2,3)$, $(-2,4,-3)$.
- Give the component form of a vector that points from $(1,2,3)$ to $(-2,4,9)$.
- Give a vector equation of the line that passes through the point $(1,2,3)$ and $(-2,4,9)$ (all distances are in meters).
- An object starts at $P=(1,2,3)$ and each unit of time its displacement is $\vec v=(-4,5,1)$. Give an equation for the position $(x,y,z)$ at any time $t$.
- What is the speed of an object that follows the path describe above?
- Use the law of cosines ($c^2=a^2+b^2-2ab\cos\theta$) to find the angle between the vectors $(-2,1)$ and $(1,3)$.
- Use the law of cosines to find the angle between the vectors $(1,2,3)$ and $(-2,4,9)$.
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