Date modified: Wed 2024-07-31 . 12:10 PM
Date created: Wed 2024-07-31 . 12:10 PM
# Week 5 Problem Set A. Due: Wednesday 10/4 ### Reading. Thomas Chapters 2.1, 2.2, 2.3 for today. And we will finish the rest of chapter 2 between this week and next week. Stewart Chapters 1.4, to 1.8. Hey you! You can do it! ### Warm-up geometry. In a square, a line segment is drawn joining from one corner to the midpoint of an opposing side. A small circle is inscribed in the triangle formed (that is, the sides of the triangle are tangent to the circle). The small circle has radius $r$. What is the length $L$ of the large square in terms of $r$? ![[1 teaching/smc-fall-2023-math-7/week-5/---files/week-5A-problems 2023-09-26 11.34.53.excalidraw.svg]] %%[[1 teaching/smc-fall-2023-math-7/week-5/---files/week-5A-problems 2023-09-26 11.34.53.excalidraw|🖋 Edit in Excalidraw]], and the [[smc-fall-2023-math-7/week-5/---files/week-5A-problems 2023-09-26 11.34.53.excalidraw.dark.svg|dark exported image]]%% ### Average rates of change. 1. In each of the following, find the average rate of change of the function over the given interval: 1. $f(x) = x^{3} +1$ over the interval $[2,3]$. 2. $f(x) = x^{3}+1$ over the interval $[-1,1]$. 3. $g(x) = 2+ \cos(x)$ over the interval $[\frac{\pi}{3}, \frac{\pi}{2}]$. 4. $r(\theta)= \sqrt{4\theta+1}$ over the interval $[0,2]$. 5. $p(t)=t^{3}-4t^{2}+5t$ over the interval $[1,2]$. ### Slope of a curve at a point and tangent lines. 1. In each of the following, find (a) the slope of the curve at the given point $P$, and (b) an equation of the tangent line at $P$. 1. $y=x^{2}-3$ at $P(2,1)$. // Typo corrected. // 2. $y=5-x^{2}$ at $P(1,4)$. 3. $y=x^{2}-4x$ at $P(1,-3)$. 4. $y=x^{3}-12x$ at $P(1,-11)$ ### Instantaneous rates of change. 1. Below shows the time-to-distance graph for a car accelerating from stationary: ![[1 teaching/smc-fall-2023-math-7/week-5/---files/Pasted image 20230926111107.png]] 1. Estimate the slopes of the secants $PQ_{1}, PQ_{2},PQ_{3},$ and $PQ_{4}$ using the figure. What are the units? 2. Using the slopes of the secants as $Q_{i}$ gets closer to $P$, estimate the instantaneous speed of the car at $t=20$ seconds. 2. Let $g(x)=\sqrt{x}$ for $x\ge 0$. 1. Find the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,2]$, $[1,1.5]$, and $[1,1+h]$, for some arbitrary $h$. 2. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for values of $h$ approaching $0$, say $h=0.1$, $h=0.01$, $h=0.001$, $h=0.0001$, $h=0.00001$, and $h=0.000001$. Use a calculator for this. 3. What does your table suggest is the instantaneous rate of change of $g(x)$ at $x=1$? 4. Now, actually calculate the limit as $h$ approaches $0$ of the rate of change of $g(x)$ over the interval $[1,1+h]$. 3. Let $\displaystyle f(x)=\frac{1}{x}$ for $x \neq 0$. 1. Find the average rate of change of $f(x)$ with respect to $x$ over the interval $[2,3]$ and $[2,T]$, for some arbitrary $T$. 2. Make a table of values of the average rate of change of $f$ with respect to $x$ over the interval $[2,T]$ for values of $T$ approaching $2$, say $T=2.1$, $h=2.01$, $h=2.001$, $T=2.0001$, $T=2.00001$, and $T=2.000001$. Use a calculator for this. 3. What does your table suggest is the instantaneous rate of change of $f(x)$ at $x=2$? 4. Now, actually calculate the limit as $T$ approaches $2$ of the rate of change of $f(x)$ over the interval $[2,T]$. ////