Unless a student has a documented excused absence, late homework will not be accepted. There are many resources available to assist you with doing your homework (e.g., office hours, course Google Group, free tutoring at numerous places across campus). You are allowed and encouraged to work together on homework. However, each student is expected to turn in his or her own work.

You are strongly encouraged to ask questions in the course Google Group, as I (and hopefully other members of the class) will post comments there for all to benefit from. You can also encouraged to stop by during my office hours and you can always email me. Lastly, free tutoring is available in AMB 137 through the Math Achievement Program.

## Daily Homework

The Daily Homework will generally consist of solving problems from the IBL course notes (PDF). On the day that a homework assignment is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the solutions/proofs that are due that day. Students are allowed (in fact, encouraged!) to modify their written solutions in light of presentations made in class; however, you are required to use the colored marker pens provided in class.

• Daily Homework 1: Read the syllabus and write down 5 important items. Note: All of the test dates only count as one item. Turn in on your own paper at the beginning of class. (Due Wednesday, 9/2)
• Daily Homework 2: Stop by my office (AMB 176) and say hello. If I’m not there, just slide note under my door saying you stopped by. (Due Friday, 9/4 by 5PM)
• Daily Homework 3: Read the Introduction and Chapter 0 up to Problem 11 of Differential Calculus (PDF) and complete Problems 3-11. In addition, skim through the Introduction and Chapter 1 of Calculus I Lecture Notes. (Due Wednesday, 9/2)
• Daily Homework 4: Read Required Knowledge from Precalculus and write down 3 things you have trouble remembering. (Due Thursday 9/3)
• Daily Homework 5: Read the rest of Chapter 0 in Differential Calculus and complete Problems 12, 14, 15, 17-19. For Problem 14, feel free to use a graphing calculator or a computer to obtain the graph of the function. (Due Friday, 9/4)
• Daily Homework 6: Read Chapter 1 in Differential Calculus up to Problem 32 and complete Problems 21-29, 31, 32. Hint: Problem 25 is hinting at tangent lines (see Definition 30). (Due Wednesday, 9/9)
• Daily Homework 7: Read the rest of Chapter 1 in Differential Calculus up to Problem 41 and complete Problems 33-35, 37-41. Hints: For Problem 34, use Problem 26 and the idea from Problem 25. For Problem 35 use the same approach as Problem 34. For Problems 37 and 38, use Problems 33 and 34, respectively. (Due Friday, September 11)
• Daily Homework 8: Read the rest of Chapter 1 and Chapter 2 in Differential Calculus up to Problem 51. In addition, complete Problems 43-45, 47-51. Note: For Problems 47-51, feel free to utilize technology to graph the functions, but you should also attempt to explore the functions algebraically. (Due Monday, September 14)
• Daily Homework 9: Read Chapter 2 in Differential Calculus up to Theorem 68. In addition, complete Problems 52-54, 56-58, 66, 67. Also, read Theorem 68 and try to draw to picture that captures its meaning. Note: For Problems 52-54, feel free to utilize technology to graph the functions, but you should also attempt to explore the functions algebraically. Hint: For the third part of Problem 67, look up the formula for factoring a difference of two cubes. (Due Wednesday, September 16)
• Daily Homework 10: Read the rest of Chapter 2. In addition, complete Problems 69-75. For Problem 69, you should just look up the answer (e.g., use a Google search). Hints: For Problem 72, you cannot “pull the 4 out of the sine function”. However, a key observation is that $4\theta\to 0$ as $\theta\to 0$. For Problem 73, consider multiplying the expression by $4/4$. For Problem 74, consider multiplying by the conjugate and using a relevant trig identity. For Problem 75, use the Sandwich/Squeeze Theorem. (Due Friday, September 17)
• Daily Homework 11: Complete Exercises 2.4.16 and 2.4.17 in Calculus I Lecture Notes. (Due Monday, September 21)
• Daily Homework 12: Complete Exercises 2.6.6-2.6.11 in Calculus I Lecture Notes. (Due Thursday, September 24)
• Daily Homework 13: Read Chapter 3 up to Problem 89 in Differential Calculus. In addition, complete Problems 78, 80, 81, 83-89. Note: For all five parts of Problem 81, you should use what you learned in Problems 78 and 80 instead of computing the derivative using Definition 76. For Theorem 82, your job is to prove both parts using Definition 76. For Theorem 87, your job is to prove the product rule. For Problem 88, rewrite $x^3$ as $x\cdot x^2$ and use your work on previous problems together with the Product Rule. Similar hint for Problem 89. (Due Wednesday, September 30)
• Daily Homework 14: Read Chapter 3 up to Problem 93. In addition, complete Problems 90, 92, 93. Also, complete the statement of Theorem 91. (Due Thursday, October 1)
• Daily Homework 15: We are going to hop around a little bit for the next few homework assignments. Complete Problems 132-134, 136(a)(b), 139, 141, 142 in Differential Calculus. Also, complete the statement of Theorems 135 and 140 along the way. (Due Friday, October 2)
• Daily Homework 16: Complete Problems 121, 125-129, 136(c), 137, 138 in Differential Calculus. Hints: To do 125 and 126, you will need Theorem 124. Also, it may be useful to know in advance that $\frac{d}{dx}[\sin(x)]=\cos(x)$ and $\frac{d}{dx}[\cos(x)]=-\sin(x).$ To compute the derivatives of the functions for Problem 137, you should use the derivatives that I just mentioned and the quotient rule. (Due Monday, October 5)
• Daily Homework 17: Complete Problems 149-153 in Differential Calculus. Hints: It will be useful to know that $\frac{d}{dx}[e^x]=e^x$ and $\frac{d}{dx}[b^x]=b^x\ln(b).$ You’ll prove that the first formula is correct in Problem 150. For now, you can just assume that the second formula is correct. (Due Wednesday, October 7)
• Daily Homework 18: Complete Problems 160, 163-166 in Differential Calculus. Hints: It will be useful to know that for $x>0$ $\frac{d}{dx}[\ln(x)]=\frac{1}{x}$ and $\frac{d}{dx}[\log_b(x)]=\frac{1}{x\ln(b)}.$ Also, when the author writes $\overline{f}$, you should just write $f^{-1}$. (Due Thursday, October 8)
• Daily Homework 19: Read Section 3.10 in Calculus I Lecture Notes. In addition, copy down the proof of Proposition 3.10.1 and make sure you understand it. Notice that the proof is using the result of Problem 164 from Differential Calculus. Then complete Exercises 3.10.2 and 3.10.3 in Calculus I Lecture Notes. Hints: For Problem 3.10.2, mimic the proof of Proposition 3.10.1. (Due Friday, October 9)
• Daily Homework 20: Complete Exercises 3.11.2 and 3.11.3 from Calculus I Lecture Notes. (Due Monday, October 12)
• Daily Homework 21: Complete any 10 problems from 15-38 in Section 3.13 of Calculus I Lecture Notes. (Due Wednesday, October 14)
• Daily Homework 22: Complete Problems 167-170, 173, 175 in Differential Calculus. Hints: The point of Problem 169 is to not just use the Power Rule because we are trying to understand why the Power Rule works for fractional exponents. Instead rewrite $y=x^{4/3}$ as $y^3=x^4$ and use implicit differentiation. Alternatively, you can try to use the log trick. To prove Theorem 170, mimic what you did in Problem 169. The point of Problem 173 is to prove the derivative formula for $b^x$. To do this, use the log trick. Prove Theorem 175 by using the log trick. (Due Wednesday, October 21)
• Daily Homework 23: Complete Problems 94, 97-100, 102-105 in Differential Calculus. Also, digest the relevant definitions nearby. (Due Thursday, October 22)
• Daily Homework 24: Complete Problems 106-113 in Differential Calculus. (Due Friday, October 23)
• Daily Homework 25: Complete Problems 114-119 in Differential Calculus. (Due Monday, October 26)
• Daily Homework 26: Complete the following exercises. (Due Wednesday, October 28)
1. Sketch the graph of a function that is continuous on $[0,4]$, has an absolute min at 1, an absolute max at 2 and a local min at 3.
2. Sketch the graph of a function on $[1,4]$ that has an absolute max but no absolute min.
3. Sketch the graph of a function on $[1,4]$ that is not continuous but has both an absolute max and an absolute min.
4. Find the absolute max and absolute min values of $f$ on the given interval. You may assume the function is continuous on the interval.
• $f(x)=3x^4-4x^3-12x^2+1$, $[-2,3]$
• $f(x)=x-\ln(x)$, $[0.5,2]$ (You may use a calculator to evaluate $x$-values after you have the critical points.)
• $f(x)=x-2\arctan(x)$, $[0,4]$
• Daily Homework 27: Complete corresponding problems on WeBWorK. (Due Monday, October 30 by 10:20am)
• Daily Homework 28: Complete Examples 2-5 on Applied Optimization handout. (Due Friday at end of class)
• Daily Homework 29: Complete the following exercises. (Due Wednesday, November 4)
1. Sketch a graph of the function with the following properties:
• $f(-4)=2$, $f(-2)=5$, $f(-1)=2$, and $f(0)=0$
• vertical asymptote at $x=3$ such that $\displaystyle \lim_{x \to 3^{-}}f(x)=-\infty$ and $\displaystyle \lim_{x \to 3^{+}}f(x)=\infty$
• horizontal asymptote at $y=0$ such that $\displaystyle \lim_{x \to \infty}f(x)=0$ and $\displaystyle \lim_{x \to -\infty}f(x)=0$
• $f’(-2)=0$ and $f’(0)=0$
• $f’(x) >0$ on $(-\infty,-2)$
• $f’(x)< 0$ on $(-2,0)$, $(0,3)$, and $(3,\infty)$
• $f^{\prime\prime}(-4)=0$, $f^{\prime\prime}(-1)=0$, and $f^{\prime\prime}(0)=0$
• $f^{\prime\prime}(x) >0$ on $(-\infty, -4)$, $(-1,0)$, and $(3,\infty)$
• $f^{\prime\prime}(x) <0$ on $(-4,-1)$ and $(0,3)$
2. Sketch the graph of the following functions by following the algorithm we discussed in class.
• $f(x) = \displaystyle \frac{x^2}{x-2}$
• $g(x) = \displaystyle xe^x$
• Daily Homework 30: Complete all 11 parts of Exercise 4.1.6 in Section 4.1 of Calculus I Lecture Notes. (Due Friday, November 6)
• Daily Homework 31: Complete Problems 179-183 in Differential Calculus. (Due Friday, November 13)
• Daily Homework 32: Complete Problems 185-197 in Differential Calculus. (Due Monday, November 16)
• Daily Homework 33: Complete Problems 198, 200-204, 206-209 in Differential Calculus. (Due Wednesday, November 18)
• Daily Homework 34: Complete Problems 211-216 in Differential Calculus. (Due Friday, November 20)
• Daily Homework 35: Complete Problems 218-227 in Differential Calculus. Your job on 224 is to come up with a definition. Hint: Check out Section 5.2 in Calculus I Lecture Notes. (Due Wednesday, November 25)
• Daily Homework 36: Complete the following. (Due Monday, November 30)
1. Consider the integral $\displaystyle \int_0^1 3x+1\ dx$.
• Compute the value of the integral using a limit of Riemann sums and right endpoints.
• Verify that your answer is correct by interpreting the integral in terms of areas of geometric shapes.
2. Compute the value of $\displaystyle \int_0^1 x^2-4x\ dx$ using a limit of Riemann sums and right endpoints.
• Daily Homework 37: Complete Problems 235-236 in Differential Calculus. Also, complete Exercises 5.6.5(5) and 5.6.6 in Section 5.6 of Calculus I Lecture Notes. Lastly, complete exercises 17-19, 35 from Section 5.8 of Calculus I Lecture Notes. (Due Wednesday, December 2)
• Daily Homework 38: Complete Problems 230-233 in Differential Calculus. (Due Thursday, December 3)
• Daily Homework 39: Complete Exercises 5.7.5 (replace $\sinh(x)$ with $\sin(x)$ on part 4), 5.7.7, and 5.7.10 in Section 5.7 of Calculus I Lecture Notes. (Due Wednesday, December 9)

## Weekly Homework

The Weekly Homework assignments are to be completed via WeBWorK, which is an online homework system. You should log in with your NAU credentials.

• Weekly Homework 1: Complete the corresponding problems on WeBWorK. (Due Wednesday, 9/9 by 8PM)
• Weekly Homework 2: Complete the corresponding problems on WeBWorK. (Due Tuesday, September 15 by 8PM)
• Weekly Homework 3: Complete the corresponding problems on WeBWorK. (Due Tuesday, September 22 by 8PM)
• Weekly Homework 4: Complete the corresponding problems on WeBWorK. (Due Tuesday, October 6 by 8PM)
• Weekly Homework 5: Complete the corresponding problems on WeBWorK. (Due Tuesday, October 13 by 8PM)
• Weekly Homework 6: Complete the corresponding problems on WeBWorK. (Due Tuesday, October 27 by 8PM)
• Weekly Homework 7: Complete the corresponding problems on WeBWorK. (Due Tuesday, November 3 by 8PM)
• Weekly Homework 8: Complete the corresponding problems on WeBWorK. (Due Tuesday, November 17 by 8PM)
• Weekly Homework 9: Complete the corresponding problems on WeBWorK. (Due Tuesday, November 24 by 8PM)
• Weekly Homework 10: Complete the corresponding problems on WeBWorK. (Due Wednesday, December 2 by 8PM)

# Dana C. Ernst

Mathematics & Teaching

Northern Arizona University
Flagstaff, AZ
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