Homework 02

Assessment: Each item is worth 1 point. The total for this homework is 25 points. There are no partial points. The best, rather than just correct, solutions are presented well and have a logical flow in the reasoning. Write the solutions for yourself, your future self, and an audience.

If you want to have your homework checked, please send an email to oer.dlsu.soe@gmail.com.

Exercise A

This exercise is for you to practice the computation of limits using the building blocks and the rules.

My suggestion is to be explicit about what rules and building blocks you are using in order to obtain your final answer. It helps build reasoning skills. After you get used to it, things become more automatic and you could choose to be less explicit.

Some guides and warning signs are also incorporated. Verify your final answers with SymPy.

  1. \(\displaystyle \lim_{y\to 0}\frac{(y+1)^5-y^5}{y+1}\)

    1. Note: You may choose to expand \((y+1)^5\) but this may be unnecessary here, but feel free to explore this alternative in order for you to build up some ability to recognize what could shorten the amount of work.
    2. Algebra issues: You may be tempted to write \[\frac{\cancel{(y+1)^5}^4-y^5}{\cancel{y+1}}=(y+1)^4-y^5\] This is INCORRECT. Do the following sanity check to convince yourself that this approach is INCORRECT: Substitute \(y=2\) (for example or try some other valid \(y\)) to the expression \(\dfrac{(y+1)^5-y^5}{y+1}\) and to \((y+1)^4-y^5\). Are they the same?

  2. \(\displaystyle \lim_{x\to 1} \frac{x^2+7x-8}{x-1}\)

    1. Note: It may be a good idea to simplify \(\dfrac{x^2+7x-8}{x-1}\) first. The numerator could be factored as a product of \((x-1)\) and another term.
    2. Note: This is another example where if you just directly substitute \(x=1\) into \(\dfrac{x^2+7x-8}{x-1}\) will not work. This is the subtle difference between computing the limit as \(x\to 1\) versus computing the function value at \(x=1\).

  3. \(\displaystyle \lim_{x\to -1} \frac{x^2+7x-8}{x-1}\)

    1. Note: Do you think it would really be necessary to simplify \[\frac{x^2+7x-8}{x-1}\] first in this situation? Or it is more of an optional move compared to the previous item?

  4. \(\displaystyle \lim_{x\to 0} \frac{x^3+3x^2-2x}{x}\)

    1. Algebra issues: You may be tempted to write \[\frac{\cancel{x^3}^2+3x^2-2x}{\cancel{x}}=x^2+3x^2-2x=4x^2-2x\] This is INCORRECT. Do a sanity check as to why this is incorrect.
    2. A less problematic algebra issue: You may find yourself writing \[\frac{\cancel{x^3}^2+3\cancel{x^2} ^1-2\cancel{x}}{x}=x^2+3x-2\] The final answer is correct, but I would suggest an improvement in the writing. Instead, write as \[\frac{x^3+3x^2-2x}{x}=\frac{\cancel{x}(x^2+3x-2)}{\cancel{x}}=x^2+3x-2\] Another way to express the idea you have seen is \[\frac{x^3+3x^2-2x}{x}=\frac{x^3}{x}+\frac{3x^2}{x}-\frac{2x}{x}\] Yet another way is to write \[\frac{x^3+3x^2-2x}{x}=\frac{1}{x}\left(x^3+3x^2-2x\right)\] and then multiply.

  5. \(\displaystyle \lim_{h\to 0}\frac{(x+h)^3-x^3}{h}\), with \(h\neq 0\)

    1. NOTE: Pay attention to what variable is going to 0. Compare with Item 6.
    2. Algebra issues: You may have to expand \((x+h)^3\) here. But be careful – \((x+h)^3\) is NOT \(x^3+h^3\). Do a sanity check to convince yourself of this. Try different combinations of \(x\) and \(h\) and explore if the both sides are equal to each other.
    3. Connection: Is there a connection to a Newton quotient here? Is there a connection to a derivative here? Explain.

  6. \(\displaystyle \lim_{x\to 0}\frac{(x+h)^3-x^3}{h}\), with \(h\neq 0\)

    1. NOTE: Pay attention to what variable is going to 0.
    2. Connection: Compare with Item 5.

  7. \(\displaystyle \lim_{h\to 2}\frac{\dfrac{1}{3}-\dfrac{2}{3h}}{h-2}\)

  8. \(\displaystyle \lim_{x\to 4}\frac{2-\sqrt{x}}{x-4}\)

    1. You may have to modify the trick found here.

Exercise B

Attempt this exercise if you have practiced Exercise A very carefully and have built enough confidence in algebra skills. There is a lot to gain for attempting this exercise, even if you get stuck and get things incorrectly.

The task is to find \[\lim_{h\to 0} \frac{\sqrt[3]{27+h}-3}{h}.\] There are many ways to do this.

  1. A first approach will be to extend the idea behind the approach here. The issue is that you have the cube root rather than the square root. Here you need to resort to some algebraic tricks. Show first that \[\left(\sqrt[3]{x}-\sqrt[3]{y}\right)\left(\sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2}\right)=x-y.\] Afterwards, focus on the numerator and do some pattern matching who should play the role of \(x\) and who should play the role of \(y\)? The idea is that \(h\) will get cancelled out later. Once this becomes clear, you would be able to compute the limit.
  2. A second approach is to use another trick. Let \(u=\sqrt[3]{27+h}\). Solve for \(h\) in terms of \(u\) and show that you can compute \[\lim_{u\to 3} \frac{u-3}{u^3-27}\] instead. Pay attention to the change of variables. You may also have to resort to an algebra trick similar to what you find in Item 1.
  3. A third approach will be to recognize that the requested limit is related to a Newton quotient. Can you recognize it? Is it connected to a derivative? If it is, what is this derivative, specifically?

Exercise C

This exercise is for you to practice the computation of derivatives using the shortcuts developed so far. A lot of the exercises would be about pattern matching and making sure you have the right patterns which enable you to apply the right shortcuts.

My suggestion is to be explicit about what rules you are using in order to obtain your final answer. It helps build reasoning skills. After you get used to it, things become more automatic and you could choose to be less explicit.

Some guides and warning signs are also incorporated.

Find the derivatives with respect to the appropriate variable for each of these functions. There is no need to simplify your answers. If you want, you can use factor() and simplify() after inputting your expression in SymPy. You can use that to help you learn how to simplify and factor expressions.

You can verify your final answers with SymPy by using the appropriate commands. If you

  1. \(f(x)=\pi^{2/3}\)

    1. NOTE: \(\pi\) here is a constant.

  2. \(g(t)=9t^{10}\)

  3. \(h(x)=\dfrac{g(x)-5}{3}\)

    1. NOTE: \(g(x)\) is some function, but you do not know its form.
    2. NOTE: You do not have to use the quotient rule here, but you feel free to do this and compare with the shorter approach.

  4. \(f(x)=\dfrac{3}{x^2}\)

    1. NOTE: You may choose to use the quotient rule here, but I think it may be too much. Try it if you want.
    2. NOTE: But consider expressing the fraction in the form \(x^m\) where \(m\) is a constant. Then, can you apply the power rule?

  5. \(r(A) = \dfrac{1}{A^2\sqrt{A}}\)

    1. NOTE: You may choose to use the quotient and product rules here, but I think it may be too much. Try it if you want.
    2. NOTE: But consider expressing the fraction in the form \(A^m\) where \(m\) is a constant. Then, can you apply the power rule?

  6. Variations on a theme:

    1. \(g(r)=Ar^{b+1}\), where \(A\) and \(b\) are constants.
    2. \(g(A)=Ar^{b+1}\), where \(r\) and \(b\) are constants.
    3. \(g(b)=Ar^{b+1}\), where \(A\) and \(r\) are constants. Can you apply the power rule here? Do we have a shortcut developed for this case already? If yes, apply it. If no, explain.

  7. \(p(x)=\left(x^5+\dfrac{1}{x}\right)\left(x^5+1\right)\)

    1. NOTE: You can choose to use the product rule here.
    2. NOTE: An alternative is to expand \(p(x)\) first, then things may look easier to handle.

  8. \(f(s)=\dfrac{\left(s^2+1\right)s^{1/2}}{s}\)

    1. NOTE: You can shorten the amount of time and effort here by simplifying the expression first before computing the derivative.

  9. \(s(t)=\dfrac{\sqrt{t}+2}{\sqrt{t}-1}\)

  10. \(h(t)=\dfrac{1}{at^2+bt+c}\), where \(a\), \(b\), and \(c\) are constants

    1. NOTE: You may be inclined to write \[\dfrac{1}{at^2+bt+c}=(at^2+bt+c)^{-1}\] This is correct algebraically. Unfortunately, you cannot use the power rule here directly. Why? You will be able to do so in the next set of recordings.
    2. NOTE: This means you may have no choice but to use some other rule.
    3. Algebra issues: You may be tempted to write \[\dfrac{1}{at^2+bt+c}=(at^2)^{-1}+(bt)^{-1}+c^{-1}\] This is INCORRECT. Try doing a sanity check by choosing values of \(a\), \(b\), \(c\), and \(t\) and check whether both sides are equal or not.

  11. \(h(s)=5e^s-3x^3+8\)

  12. \(f(x)=(x+e^x)^2\)

  13. \(g(t)=(1+t^p) e^t\), where \(p\) is a constant

Exercise D

This exercise combines the skills you hopefully picked up for Weeks 01 and 02.

Find the equation of the line tangent to the curve \[f(x)=\frac{x^4+1}{(x^2+1)(x+3)}\] at \(x=0\).