Overview

Here is the course website, and you can find a copy of the syllabus here. My office hours are currently:
Monday 3-4pm, East Hall Atrium
Tuesday 2-3pm, Mathlab East Hall B860
Wednesday 3-4pm, East Hall Atrium

Daily Summary

1/3: Section 1.1
Reminders about functions, linear functions, rate of change, increasing and decreasing functions.
1.1 #34, #55, #13, #35-38, #45, #67

1/5: Section 1.2
Took the student guide quiz and organized team homework groups. Review of exponential functions.
1.2 #5-8, #19, #40, #42 (find the doubling time).

1/8: Section 1.3
Discussed concavity. Some good questions to check understanding
Q: Let P(t) = P_0e^(kt). For what value of P_0, k is P(t) concave up and concave down?
Q: Same but for P(t) = P_0a^(t)
Discussed transformations of graphs, how to compose them and reviewed symmetric functions. A possible review sheet is here. Discussed invertible functions, how to find and graph inverses.
1.3 #4-5, and q(t) = m(2t+2), #33, #29-31, #58

1/10: Section 1.4
Finished 1.3 by discussing composition of functions.
1.3 #10, #67-70, #49, #51
Introduced log as the inverse of 10^x and ln as the inverse of e^x. Discussed relationship between log rules and exponent rules.
1.4 #41, #51, #55, Math 105 F 2016 Exam 1 #1, #8

1/12: Section 1.5
Quiz on 115 2015 W Exam 1 #8, 9.
Review of Trig functions, and more extensive discussion of arccos.
1.5 #31 (Hint: first solve 1 = 8cos(2t) - 3)
Not assigned, but good review: 115 2015 W Exam 1 #6

1/17: Section 1.6
Discussion of power functions, polynomials including how to find their limiting behavior, zeroes with multiplicities, and graphs.
1.6 #12-14, #17, #18, #23
Graph y = x^3-x, y = (x^2 + 2x+1)(1-x)^2

1/19: Section 1.7
Gave a quiz on 115 2015 W Exam 1 #6 and a few extra problems .
Gave a discussion on rational functions including how to find their zeroes, asymptotes, long-term behavior and how to use this information to graph them. Introduced the idea of continuous functions.
Graph y = 1/(x^2-1), y = (x^2 - 5x + 6)/(x+2)^2, y = (x^3 - x)/(x-2). 1.7 #1

1/22: Section 1.7 and 1.8
Discussion of one-sided limits, limits, and continuous functions in terms of limits. Examples and explanations of how to find limits algebraically and using graphs
1.8 #2
Let f(x) = 1 if x < 1, and 0 else. Let g(x) = cos(x). What are the right hand and left hand limits at zero of f(g(x))? (Hint: They are both 1. Why?)
1.7 #24, 28 1.8 #38
Find the limit at zero of p(t) = (1-cos(t))/t^2 (Use your calculator)
1.7 #33, 36 1.8 #57, 45, 50

1/24: Section 2.1 and 2.2
Discussed algebraic properties of limits. Combining continuous functions in various ways gives continuous functions. Moved on to discuss average velocity, and its interpretation as the slope of secant line, and how to take a limit to get instantaneous velocity, and its interpretation as a tangent line at a point. Defined the derivative at a point, and gave examples of how to compute the derivative at a point using limits.
1.8 #9, 2.1 #10, 24, 27

1/26: Section 2.2
Gave a quiz (last problem to be handed out on 1/29). Review of 2.2. Brief introduction to 2.3, the derivative function.
Estimate the derivative of f(x) = x^x at x=2 using your calculator and a table. (Hint: you'll need to know the limit definition of the derivative for this one - derivative rules from AP calculus won't work)
2.2 #58, 22

1/29: Section 2.3
Defined the derivative function and gave various interpretations in terms of the slope of the tangent line at a point, and when the deriviative is positive and negative based on if the original function is increasing or decreasing. Also reviewed how to figure out from a pair of functions which is the original function, and the derivative. How to estimate the derivative from a table.
2.3 #10, 42-47, 56, 52
Sketch the derivative of f(x) = cos(x) on [-2pi, 2pi] (Sketch using the graph - forget what you know about derivative rules for a moment)
W 2012 Exam #1 Problem #2 (last problem)

1/31: Section 2.4
Gave a new interpretation of the derivative at a point as an approximation of the function near that point.
2.4 #1
Old Exam questions
Solutions

2/2: Section 2.5
Gave a quiz, and discussed the solutions.
Introduced the second derivative as the derivative of the first derivative. Explained how to interpret the second derivative in terms of the original function, i.e. how the slopes of the tangent lines are changing. Described how the sign of the second derivative determines the concavity of the function.
2.5 #8-13, #4, 5, 37

2/5: Review Day
Reviewed how to interpret equations involving the derivative of an inverse function, then went over several questions from Fall 2017 Exam 1.
Fall 2017 Exam 1.
Solutions.

2/7: Section 2.6 and 3.1
Discussed what it means for a function to be differentiable and gave three examples how a function can fail to be differentiable by being either; not continuous, having a sharp corner/cusp or if the slope of the tangent line is undefined. Also gave an example of a function that is continuous everywhere, but differentiable nowhere. Went on to discuss the power rule, and how you can use it to compute the derivatives of polynomials.
2.6 #11, 13
Let g(x) = ax + 2 if x < 0, b(x-1)^2 if x >= 0. Find a, b so that g is continuous and differentiable everywhere.
Let p(t) = 4(t-2)^7. Compute p'(t). Notice that you cannot directly apply the power rule here. (Hint: transform the function so you can apply the power rule. How do you then find the derivative of the original function?)
Compute the derivative of f(x) = x^3 using limits (not using the power rule).

2/9: Section 3.2
Gave a quiz. Discussed how f(x) = e^x is its own derivative, and gave the general formula for the derivative of exponential functions. Sketched the derivative of cosin, then gave formulas for the derivative of trigonometric functions.
3.2 #29, 34, 43
2.5 #2, 50
If f(x) = sin(x), what is the 98th derivative of f?

2/12: Section 3.3, Intro to 3.4
Gave a brief discussion how you could guess the product rule. Presented the product rule and the quotient rule. Brief review of decomposing functions in preparation of the chain rule.
3.3 #3, 7, 31, 18, 44, 50
Let p(x) = f(x)g(x)h(x). Find a formula for p'(x) in terms of f, g, h, f', g' and h'.
Decompose r(t) = cos^2(t) and v(s) = tan(e^(s^(1/2)))

2/14: Section 3.4
Discussed how one could guess the chain rule. Gave the formula for the chain rule.
3.4 #2, 36 89, 91
Derive the quotient rule using the product rule and the chain rule (Hint: p(x)/q(x) = p(x)q(x)^(-1))
Fall 2016 Exam 2 #1
Find the derivative of f(x) = xsin^2(x) + xcos^2(x).

2/16: Section 3.6
Gave a quiz on W 2017 Exam 2, #4 with a few extra problems . Discussed how to find the derivative of ln(x) using the derivative of e^x, and more generally how to find the derivative of f^(-1) using the derivative of f. Gave derivative rules for ln, arctan, arcsin, arccos.
3.6 #11, 45

2/19: Section 3.7
Discussed implicit differentiation.
3.7 #2, 3, 16

2/21: Section 3.9 + Quadratic Approximation
Finished discussion of implicit differentiation. Review of the linear approximation of a function using the first derivative, and gave the definition of the quadratic approximation of a function.
3.7 #34
Winter 2017 Exam #2 #9
3.9 #5, 20

2/23: Quadratic Approximation
Gave a quiz on Fall 2017 Exam 2 #1, #8. Discussed how the sign of the second derivative can tell you if a linear approximation is an under or over estimate. More examples of quadratic approximation.
Find the quadratic approximation for
1) f(x) = e^x/(x+1) near x = 1
2) p(t) = sin(pi*t) near t = 0
3) v(s) = 4s^2 + s - 1 near s = -3. Expand your answer (It should look familiar)
If f is concave down, and near x = 1, f(x) ~= -2 + (1/3)(x-1), which of these is possible? Why or why not.
a) f(5) = 0
b) f(10) = 0
c) f(10) = 0 and f(12) = 0

3/5:The Mean Value Theorem and Section 4.1
Explained the mean value theorem, and gave examples of when the hypotheses might not necessarily hold, but the conclusion still holds. Moved on to local extrema and how to find them using critical points and the first derivative test.
3.10 #6-9, 10
4.1 #29
Find all critical points and local extrema for f(x) = (x)^(1/3)(x^2+1)

3/7: Section 4.1 and 4.2
An alternate way to find local extrema using the first derivative test, and inflection points, which are related to local extrema. Then discussed the Extreme Value Theorem for how to find global maxima on a closed interval.
4.1 #21, 16
4.2 #6, 12

3/9: Review
Gave a quiz on Fall 2016 Exam #2, #6, 7
Reviewed the Extreme Value Theorem with Fall 2016 Exam #2, #8
Gave an introduction to section 4.3, optimization with modeling.
4.3 #1 - Try to guess the correct answer with no calculus, and justify your guess, again with no calculus.

3/12: Section 4.3
Started off by giving more examples of how to find global extrema on a non-closed interval. Moved on to optimization modeling problems.
Find the global extrema of f(x) = x(x-4)^(4/5)e^(-x)
Winter 2017 Exam 2 #8

3/14: Section 4.3, Review
More examples from section 4.3. Review with Fall 2016 Exam 2 #10.

3/17: Review
Gave a quiz on Fall 2017 Exam #2, #3 and Winter 2017 Exam #2 #10. Handed out a review packet containing
Fall 2017 Exam #2, #4 - 7
Winter 2017 Exam #2, #3, 7

3/19: Review
Completed the packet plus Fall 2017 Exam #2 #9. Reviewed the relationship between critical points, local extrema and inflection points and where f, f', f'' are increasing, decreasing, positive or negative. Reviewed how to identity the original function and its derivative given a graph of two functions.

3/21: Section 4.4
Discussion of how to find critical points and local and global extrema for families of functions depending on a parameter.
4.4 #2
Worksheet

3/23: Section 4.5
Gave a quiz. Introduced the cost, revenue and profit functions. Showed that the maximizal profit (usually) occurs when marginal cost equals marginal revenue using ideas from 4.1, 4.2, 4.3.
4.5 #4, 7, 11, 21, 36

3/26: Section 4.6
Related rates problems. Began by discussing how to differentiate an expression involving several function with respect to a variable, with some graphical explanations of what's going on. Moved on to how to solve problems involving related rates.
4.6 #3, 7, 10

3/28: Review
Fall 2016 Final #2, Winter 2017 Final #10.
Suppose a particle is moving at a constant velocity c on a trajectory given by y = x^2. Suppose it is moving right at 2 ft/sec.
1) At the point (2, 4), how fast is the particly moving in the y-direction?
2) At the point (2, 4), how fast is it moving away from the point (0, 1)?
3) What is the particles minimum distance from the point (1, 2)? (Hint: minimize distance squared, not the distance)

3/30: Section 5.1
Gave a quiz on Winter 2017 Final #1, 4, 5
Motivated why you might want to compute the area under the curve as a means of computing total change, e.g. figuring out total displacement given velocity. Gave examples of how to approximate this area using left and right Riemann sums
5.1 #2, 3

4/2: Section 5.1 and 5.2
Described how estimate your error from the process of using right and left riemann sums. Formally defined the notion of area under a curve as a definite integral.
5.1 #14 estimate your error, 20, 39.

4/4: Section 5.2 and 5.3
Handed out a worksheet on definite integrals. Stated the fundamental theorem of calculus, gave a few ways to interpret it, and showed how it can be used to compute areas under curves exactly.
5.3 #18, 35 (exact value)

4/6: Section 5.3
More examples using the fundamental theorem of calculus.
5.3 #5, 6
Worksheet

4/9: Section 5.4
Gave a quiz on Fall 2017 Final #3, 4. Listed and explained several important properties of definite integrals. Defined the average value of a function, and gave some explanation for the definition and why it's meaningful.
Worksheet 1
Worksheet 2