Date modified: Sat 2023-07-08 . 10:20 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 3 Thursday 7/6 brief notes. Today you took Exam 1. Whew! Ok, here are some commentaries. General comments. - Review basic differentiation rules - Pay attention to basic algebra - Most of you have good hand writings and organization, so keep that up! - Generally, there is a $+C$ for an indefinite antiderivative. (Recall antiderivatives are unique up to an additive constant) - When doing integration by parts, $\int udv=uv-\int vdu$, you need to subtract the entire $\int vdu$. Quite often easily forgotten the negative sign is distributed to the whole second term. Problem 1. - Careful when taking derivative (review/apply quotient rule carefully). Watch out for algebra when simplifying! - Most of you got how to relate $(f^{-1})'(b)= \frac{1}{f'(a)}$ if $f^{-1}(b)=a$. Good job. - Solving for $f^{-1}(x)$ explicitly: There is a square root, try grouping like terms when you solve. - Also, $(a+\sqrt{b})^{2}$ is **not** $a^{2}+b$. Careful\! Problem 2. - Good, this is recognized to be an indeterminant form, so we can try L'Hospital rule. - Watch out for differentiation again. Chain rule. Pay attention to the sign! - And when it is no longer an indeterminant form, you can evaluate the limit. Problem 3. - Apply integration by parts carefully. Identify what is $u$ and what is $dv$. (And then from $dv$ you find out what $v$ is) - Know the derivative/antiderivatives of the hyperbolic functions and trigonometric functions. - Pay attention to chain rule when finding antiderivatives! - Purposely this problem has fractional factors. Pay attention to algebra! - This problem is one of those where you have to apply integration by parts several times, and identify the original integral somewhere, and solve for it. - Tabular integration also works here, if apply correctly. Problem 4. - Before applying partial fractions, you should do long division here first. - The goal is once we have a rational function where the numerator has less degree than the denominator, we can perform partial fraction decomposition. The decomposed fractions are easier to deal with. - Careful with algebra again. Problem 5. - We have a quadratic here, complete the square to deal with it. - This is a "digimon" problem as called in class (Oh boy, what am I even saying.) The type of problems involving $\sqrt{1+u^{2}}$, $\sqrt{1-u^{2}}$ or $\sqrt{u^{2}-1}$. - Here you would need to deal with $\int \frac{1}{\sqrt{1+u^{2}}}du$. This is not $\arctan(u)$. Recall $\int \frac{1}{1+u^{2}}du=\arctan(u)+C$. - Try an appropriate trigonometric substitution. - Some of you skirt around the problem of "express final result without any trigonometric functions" by recognizing that we can actually use an appropriate inverse hyperbolic function. That's fine, you found a technicality loop hole. Just not intended but still ok. - Regardless, you still need to pay attention to chain rule correctly if used an inverse hyperbolic funciton. - Algebra: Careful, $\sqrt{a^{2}+b^{2}}$ is **not** $a+b$ ! - If you do it the trigonometric substitution way, you will need $\int\sec(x)dx$. We have derived this in class as formula (and homework). - Once you finished the trigonometric substitution, use a right triangle to figure out what each piece is. Problem 6. - This is a rational function in $\sin(x)$. What technique can we use to figure this one out? Starts with "W"... - Once you found the technique, do algebra correctly. - And integrate correctly. - There is actually another way of doing this problem. Multiply top and bottom by something clever. Some of you discovered it, good job! (You will then need to finish the integration, each part is not bad) Problem 7. - This one is integration by parts. - One of the parts requires to integrate some trigonometric function. - And you will need to integrate some trigonometric function again. - A correct final answer would involve the terms $\sin(x),x\cos(x),\sin^{3}(x),x\cos^{3}(x)$. - You can use tabular integration to help organize your work. - You can do it! Problem 8. - Note. If there is an exception to the statement, then the statement is **false** and not true! - For these, review your notes and homework. - Find counterexamples or explain why the correct answer is so.