Below, you can check your homework answers after you have attempted the problem yourself. Of course, your submission should include a full explanation and not just the final answer. The purpose of this page is only to give you feedback as quickly as possible to see if you need to review your work or see me or QUAILs at student hours or CLEAR tutoring sessions. The buttons will ask you to click twice to confirm you want to show the answer.
3.6.238:\(f(1)=\pm\sqrt{\frac{10}{12}}\) or \(f(1)=\frac{10}{8}\). This question received errata changing the cube to a square, and the PDF version of the textbook doesn't have the update.
3.6.239:\(f'(-1)=\frac{43}{4}\)
3.6.253(a):\(\approx-0.5831m/s\)
3.6.253(b):\(\approx0.2499m/s^2\)
3.6.253(c):It's slowing down (think about the relationship between the sign of the first and second derivatives)
Additional Problem 2 (ii):\(\frac{1}{y}\frac{dy}{dx}=\ln(x)+1\)
Additional Problem 2 (iii):\(\frac{dy}{dx}=x^x\ln(x)+x^x\)
Problem Set 7
4.1.5:\(2\sqrt{3}\)ft/s
4.1.7:390mph
4.1.20:\(1600\pi m^3/s\)
4.1.30:\(\frac23\)m/min
4.2.50:\(L(x)=x\)
4.2.52:\(L(x)=2x-\frac{\pi}{2}+1\)
4.2.68:\(dy=(12x^3+2x-2)dx\)
4.2.69:\(dy=(\cos(x)-x\sin(x))dx\)
4.2.84:\(\Delta V\approx\pm 31.42mm^3\)
Additional Problem 1(a):\(L(x)=1\)
Additional Problem 1(b):\(Q(x)=1-\frac{1}{2}x^2\)
Additional Problem 1(c):Your graph should look like this Desmos page
Problem Set 8
Additional Problem 1:There are many, many possible answers, but this Desmos page has the three I thought of
4.3.108:\(x=\pm\frac{1}{2}\)
4.3.110:\(x=1\)
4.3.112:\(x=\frac{\pi}{2}+k\pi\) for any integer \(k\)
4.3.118:Global max of 19 at \(x=4\) and global min of 3 at \(x=0\)
4.3.120:Global max of 4 at \(x=-1\) and global min of 0 at \(x=0,1\)
4.3.122:Global max of \(\sqrt{8}\) at \(x=1\) and global min of 0 at \(x=9\)
4.5.198:Many possible solutions, but if you're getting stuck, draw some discontinuities where the derivative is positive on one side but negative on the other.
4.5.225c:Concave up \((2,\infty)\), Concave down \((-\infty,2)\) * Your explanation should include taking a derivative!
4.5.225d:Inflection point \(x=2\) * Your explanation should include taking a derivative!
4.6.252:\(x=-3,2\)
4.6.254:\(x=0,1,2\)
4.6.256:Plugging in \(a=-1\) gives an indeterminate form, so you should factor and cancel terms from the top and bottom. At the end, you should get a finite limit at -1, so there is no asymptote.
4.6.260:Plugging in \(a=1\) gives a finite value, so there is no asymptote.
4.6.294:Some data you should compute and take into consideration—local min at \(x=-1/3\), always concave up, end behavior goes to \(\infty\) on both sides.
Problem Set 9
4.6.262:\(\frac{1}{2}\)
4.6.263:\(\infty\)
4.6.271:Vertical asymptote at \(x=0\) and no horizontal asymptotes
4.6.273:Vertical asymptotes at \(x=\pm2\) and no horizontal asymptotes
4.6.285:Many possible answers, but your function should have a denominator that's zero when \(x=1\) and the limit as \(x\to\pm\infty\) should be 2.
4.6.287:Many possible answers, but your function should have a denominator that's zero when \(x=-1\) and the limit as \(x\to\pm\infty\) should be 4.
4.7.316:The box with the largest volume is about \(0.423\times1.174\times3.174\) meters
4.7.320:The enclosure with the largest area is \(200\times400\) feet
4.7.322:For the shortest length, the wire should be anchored 10 feet from the shorter pole (called the "second" pole in the problem for some reason 🙄)
4.7.324:90bpm
4.7.325:84bpm
4.7.333:The profit is maximized with four pizzas
Problem Set 10
4.10.466:Take the derivative of \(F(x)\) and compare with \(f(x)\)
4.10.467:Take the derivative of \(F(x)\) and compare with \(f(x)\)
4.10.470:\(F(x)=-x^{-1}+\frac12x^2+c\)
4.10.472:\(F(x)=e^x+\frac32x^2-\frac13x^3+c\)
AP1a:You should have 6(\(=\frac{12}{2}\)) pairs that sum to 13.
AP1b(i):This is a property we talked about in class—you can sum from 1 to 12, then sum from 13 to 13 (that is, just add 13 on at the end)
AP1b(ii):A good way to see why it works in general is to first write 13 as 12+1, then find a common denominator.
