Course:MATH110/Archive/2010-2011/003/Midterms and Finals

There will be two midterms in this course, one in each term, each worth 10% of the final grad. There will be two finals: the Winter final exam, worth 20% and the Spring final exam, worth 25%.

Spring Final Exam Information

The Spring Final Exam is on April 21 at 8:30am, location: OSBO A.

Content

The Spring Final Exam is cumulative and will cover the whole course with of course a stronger emphasis on more advanced applications than on technical computations.

You can use these learning objectives to check your own progress and construct your own review sessions (since you want to spend time practicing what you're not good at, not doing 20 problems on a topic you actually understand fairly well).

The learning objectives covered by this Final Exam are listed below by topics.

• LIMITS. You should be able to:
  • Explain what it means to compute the limit of a function at a specific point or at infinity.
  • Distinguish between limits on the left, on the right and how they relate to what we call THE limit.
  • Compute limits of functions (possibly using L'Hospital Rule).
• CONTINUITY
  • Explain what it means for a function to be continuous at a point in both a formal way (using limits) and a geometrical way (the intuitive explanation).
  • Decide whether or not a function is continuous at a given point from the definition.
  • Describe the main properties of continuous functions (the Intermediate Value Theorem & the Extreme Value Theorem).
• DERIVATIVES. You should be able to:
  • Give the precise definition of the derivative of a function.
  • Compute the derivative of very simple functions from that definition.
  • Describe and use the various rules (product rule, quotient rule, chain rule, etc.) which allow us to compute directly the derivative of more complicated functions.
  • Compute implicit derivatives.
  • Give a geometric illustration of what the derivative of a function tells us about the original function.
  • Describe how the Mean Value Theorem and what it tells us about differentiable functions.
• PROPERTIES OF FUNCTIONS. You should be able to:
  • Explain what the domain of a function is and compute it for a given function.
  • Describe what it means for a function to have a vertical or horizontal asymptote and link this to the notion of a limit.
  • Compute the vertical and horizontal asymptotes of a given function.
  • Compute the intervals of increase and decrease of a function.
  • Compute the intervals at which a function is concave up or concave down.
  • Compute extremas of a function, distinguishing between local max/min and global max/min.
  • Compute inflexion points.
  • Sketch the graph of a function given the above information.
• APPLICATIONS. You should be able to:
  • Construct a function which will model a specific situation given specific properties it should satisfy. (This talks about all the exponential/logarithmic growth/decay, information about instantaneous rate of changes and so on when modelling populations, interest payments, temperature, etc.)
  • Explain the idea of linear approximation of a function and compute the linear approximation of a given function at a given point.
  • Explain the idea of optimization and solve such problems.
  • Solve related rates problems.

(If you have questions about those learning objectives or want to suggest a rephrasing or such, please feel free to email me!)

Second Midterm Information

Midterm 2 is on February 9, from 6:00pm to 7:30pm, location: MATH 100

Content

The content covered on this second midterm will be

• Exponential and Log functions
• Modeling, rates of change
• Related rates and the use of the chain rule
• Linear approximation

You can get some more practice by studying the following sections of the textbook:

• Section 1.3 will describe the basics on exponentials and logarithms. Clearly, we haven't discussed much about the idea of inverse function, so we won't ask it, but it does help understand more on the relationship between exponentials and logarithms.
• Section 3.5 describes the derivative as a rate of change, this is a fundamental concept that has been explored so far. Check this section out if you're still unclear on how the derivative can be thought as a speed, a growth rate or a marginal cost.
• Section 3.8 describes the derivatives of logarithmic and exponential functions. You should be able to take the derivative of any function involving logarithms and exponentials fairly quickly and effortlessly, time to review those derivation rules. We haven't discussed logarithmic differentiation yet, you can leave it for now.
• Section 3.10 related rates describe those problems where several things depend on each other. For example, the volume of a sphere depends on its radius, which might depend on time (if we're thinking of a sphere in expansion). Then using the chain rule, we can know the rate of change of volume with respect to time if we know the rate of change of volume with respect to its radius and the rate of change of radius with respect to time. We've worked on several problems of this type, you'll find more of them in this section. Watch out, some of them are only tricky because of the underlying geometry, the calculus part is always fairly straight forward. Recall also that we haven't worked with implicit differentiation yet, so some problems might not be feasible.
• Section 4.5 describes linear approximations. Recall that in this setting, we are trying to ask questions about how to actually compute stuff, so forget about your calculator, it really doesn't exist anymore in this context. We haven't discussed differentials forget about them.

Review Questions

Try to think about these review questions and see how much you can explain. If you're unclear about a question, you can always check the appropriate section (in brackets) in the textbook to find more material to work on.

• (1.3) Explain how the property $b^{u+v}=b^{u}b^{v}$ is related to the property $\log _{b}(uv)=\log _{b}(u)+\log _{b}(v)$.
• (1.3) Given any positive number $b$, explain how to rewrite $b^{x}$ as $e^{kx}$ for some value of $k$. Conversely, starting with $e^{kx}$, how to rewrite it as $b^{x}$.
• (3.5) Describe the velocity of an object which is moving along a line at a constant negative acceleration.
• (3.5) What is the difference between the average rate of change and the instantaneous rate of change of a function?
• (3.8) Explain why $b^{x}=e^{x\ln(b)}$.
• (3.10) If two opposite sides of a rectangle are increasing in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?
• (4.5) Sketch the graph of a smooth function $f$ and label a point ($a$, $f(a)$) on the curve. Draw the line that represents the linear approximation of $f$ at that point and write its equation.
• (4.5) How do linear approximations relate to the marginal cost?

Suggested Problems

If you're looking for problems from the textbook to work on, you can always check the review questions and match them to problems in the textbook.

Here's a list of problems suggested by Dr. Leung:

• (1.3) 41-48, 58, 67
• (3.5) 2, 3, 11-14, 17, 18, 26-29, 43, 44
• (3.8) 9-20, 25, 33-36, 43-50
• (3.10) 5-12, 15-23, 26-31
• (4.5) 7-22

Winter Final Exam Information

The Winter final exame is scheduled for December 15, from 3:30pm to 6:00pm in room OSBO A (that's the Robert F. Osborne Centre). This final will examine all the material that has been studied during the first term.

• My office hours before the final will be the following:
  • Monday December 6: from 9am to 12pm.
  • Tuesday December 7: from 9am to 10am.
  • Thursday December 9: from 9am to 12pm.
  • Tuesday December 14: from 9am to 12pm.

Some review questions

• Am I more confortable with the Basic Skills? (check out each others group projects, some contains some review questions)
• What is a limit?
• How do we use limits to define the continuity of a function at a point?
• What does it mean for a function to be continuous on an interval?
• What is the Intermediate Value Theorem? What does it say? Can I illustrate it with an example and a counter-example?
• What is the derivative of a function at a point?
• How do we use limits to compute the derivative of a function at a point?
• What are the rules of differentiations and how to use them to compute derivatives without having to use limits?
• How to write the equation of the tangent line of a function at a point?
• How does the derivative of a function link to the growth (increasing or decreasing) of the original function?
• How to sketch the graph of a function? This includes sign tables, asymptotes (horizontal and vertical only)

Remarks on the material

All the material that was not covered in class but mentioned in the learning goals is of course due next term and not for this midterm. Hence:

• Inverse trigonometric functions are NOT included in the Winter Final,
• Logarithmic and exponential functions and their respective derivatives are NOT included in the Winter Final,
• L'Hospital Rule is NOT included in the Winter Final,
• Implicit differentiation is NOT included in the Winter Final.

First Midterm Information

The first midterm is scheduled for October 20, from 6:00pm to 7:30pm in room HENN 201. The material covered is as follow:

• Computational questions such as those you worked on in the WebWork assignments.
• Longer problems for which you must provide a detailed solution (and not only an answer) similar to those you worked on in the homework.
• Conceptual questions as discussed in the lectures.

Here's last year's midterm if you want to practice some questions. But remember the content of the course changes slightly each year.

More precisely, topics will be:

• basic geometry
• algebra
• functions
• limits
• continuity

Midterm Results

Here are the results from the first midterm.

File:MATH110 003 midterm 1 stats.pdf

Some review questions

• What is a rational function? How can I compute the domain of a rational function?
• What is a difference of squares? How can I simplify expressions involving one?
• How do I determine the domain of functions involving square roots?
• How can I find the intersection of two curves y=f(x), y=g(x)?
• How can I compare two rational functions with different denominators?
• What is function composition? If $f(x)=x^{2}$ and $g(x)=2x$, what is $f\circ g(x)$? What is $g\circ f(x)$?
• How do I compute the limit at infinity of a function?
• What is the the absolute value function? What does ${\frac {|x|}{x}}$ look like? What is the limit of this as $x\to 0$?
• Can I determine the value of a function at a point given its graph?
• Can I find the left and right limits of a function at a point by looking at its graph?
• What is the "error and tolerance" definition of continuity?
• What is the intermediate value theorem (IVT)? How can I use it to tell if a function has a root? What conditions on the function are required?
• In what ways can a piecewise function of continuous pieces fail to be continuous?
• What is a removable discontinuity? How can I tell without looking at a graph?