Date modified: Fri 2023-06-23 . 07:22 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 1 Wednesday Problems Reading: Finish reading chapter 6.1, 6.2$\ast$ and $6.3\ast$. And preview 6.4$\ast$, 6.6, 6.7, 6.8. Hey you, you can do it! Work together, don't give up! ## Problems. 1. Recall the properties of logarithms we learned today, do the following: 1. Re-express the following expression as one **single** logarithm:$$ 3\ln(x+5) +\frac{1}{2}\ln(x^2+x+8) - 7\ln(5-x) $$ 2. If $f(x) = \ln(x+ \ln(x))$, find $f'(1)$. 2. Consider the function $\displaystyle f(x)=\int_{-5}^{x} \frac{1}{\sqrt{t^{2}+3}}\,dt$ 1. Show this function $f$ is one-to-one. Show your work. 2. Find $f^{-1}(0)$. 3. Find $(f^{-1})'(0)$. 3. Consider $f(x)=3x + 2\ln(x)$ defined on $x > 0$. 1. Does $f$ have an inverse? Justify your answer. 2. Denote the inverse of $f$ as $g$. Find $g(3)$ and $g'(3)$. 4. Consider the function $f(x)=x^{3}+x^{2}+x+1$, defined on all of $\mathbb R$. 1. Show this function is one-to-one. Show your work. (Hint, after you differentiate, you need decide what the sign is of some parabola. **Completing the square** may help.) 2. Find $f^{-1}(1)$ and $(f^{-1})'(1)$. 3. Find $f^{-1}(0)$ and $(f^{-1})'(0)$. 4. What is the domain and range of the function $g(x)=\sqrt{x^{3}+x^{2}+x+1}$? (Hint: Use the information you gathered from the previous parts of this problem to help you, without using technology, what would a graph of $f$ look like...) 5. Recall the bounds that we mentioned for $\ln(n)$ in class, when $n\ge 2$ is an integer:$$ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} < \ln(n) < 1 + \frac{1}{2} + \cdots + \frac{1}{n-1} $$ 1. Briefly explain why this is true again graphically using the definition of $\ln(n)$. 2. Next, show the difference between the **upper bound** and the **lower bound** given above is no greater than $1$, regardless of $n$. (Take the difference!) 3. Explain why if we use the **average** of these two bounds, we are always within $0.5$ of the true value of $\ln(n)$. Hint. If two numbers $a$ and $b$ are within one unit of each other, what number would you guess so that it is always within $0.5$ away from any number between $a$ and $b$? Maybe draw on a number line to help you think about. 4. Estimate $\ln(6)$ to within 0.5, without any technology by using the previous part. Now compare it with the calculator value of $\ln(6)$. (Ok, I know this is a bit tedious by hand, but I wanted to show you a taste of the ancient world...and it's just adding fractions.) 6. We used this fact several times in class today, **the uniqueness of antiderivatives up to a constant**: If $f'(x)=g'(x)$ on some interval $I$, then $f(x) = g(x) + C$ on $I$, for some constant $C$. Let's see how far we can push this idea. 1. Suppose $f'(x) = g'(x) + 3$ on some interval $I$, what can you say about $f(x)$ and $g(x)$? Hint: Try to re-express the RHS as the derivative of one single function.... 2. Suppose $f'(x) = g'(x) + x$ on some interval $I$, what can you say about $f(x)$ and $g(x)$? 3. Suppose $f''(x) = g''(x)$ on some interval $I$, what can you say about $f(x)$ and $g(x)$? 4. Suppose $f'''(x) = g'''(x)$ on some interval $I$, what can you say about $f(x)$ and $g(x)$? 7. Using FTC, we can "easily" construct functions with a desired derivative, by using some **definite integral**. Though some attention needs to be paid on its desired domain. Let $f(x)= \frac{1}{1-x^2}$, construct the following: 1. Construct a function $g(x)$ whose derivative is $f(x)$ on the interval $(1,\infty)$. 2. Construct a function $g(x)$ whose derivative is $f(x)$ on the interval $(-1,1)$. 3. Construct a function $g(x)$ whose derivative is $f(x)$ on the interval $(-\infty,-1)$. ///