Course:MATH110/Archive/2010-2011/003/Practice Problems

Practice midterms for Spring Final Exam

Here are two practice midterms for the final exam.

• first practice midterm and its solution

You can ignore question 1a, we don't expect you to know the functions $\sec(x),\csc(x),cot(x)$.

You can also skip question 9b as we haven't studied inverse trigonometric functions.

• second practice midterm

I've heard that there are some tough limits to compute somewhere, so don't worry about it :)

General practice problems from the textbook

These questions are drawn from the Calculus: Early Transcendentals text. It is recommended that you do all of them. Solutions to odd-numbered questions can be found at the back of the text. Extra review problems can be found at the end of each chapter. You should also review your course notes, assignments, and midterm. Notice that some sections correspond to material covered in the first term. This is fair game on the final exam, though the exam will focus on material covered in the second term.

• From section 2.2: 1-5, 7-10, 21-23
• From section 2.3: 1-10
• From section 2.4: 1-6, 33
• From section 2.5: 1-8, 38-40
• From section 2.6: 1,2, 19-24, 51-56
• From section 3.1: 2-9, 49-52
• From section 3.2: 3-5, 35-41
• From section 3.3: 7-12, 17-22, 27-30, 45
• From section 3.4: 15-25, 50-55
• From section 3.5: 11-18, 25-30, 35, 43, 44
• From section 3.6: 7-30
• From section 3.8: 9-22, 24-36, 56-60, 80
• From section 3.10: 5-13, 16-23, 26-31, 45
• From section 4.1: 19-22, 31-38, 68, 73
• From section 4.2: 1-7, 11-26, 31-37, 57-62, 67-68, 80, 83
• From section 4.3: 7-26, 42-47, 51-54
• From section 4.4: 5-13, 15, 18-20, 22-23, 42, 50, 51
• From section 4.5: 1,2, 13-22
• From section 4.6: 1-6, 29-31
• From section 4.7: 13-29, 34, 36, 37, 39, 40, 44

Practice problems by topics

Here are some practice problems organized by topics.

Mean Value Theorem

Question: If the total world population was 2 of billion people in 1920 and of 6 billion in 2000; assuming that the function describing the population over time is continuous and differentiable, which of the following statements is a direct consequence of the Mean Value Theorem?

A) There is at least one precise time at which the average rate of change of the population was of 50 million persons per year.

B) The average population between 1920 and 2000 is 4 billion people.

C) The average rate of change of population between 1920 and 2000 is 50 million people per year.

D) At some point in time the population was exactly 4 billion people.:E) The rate of change of population was of 50 million persons per year at at least one point between 1920 and 2000.

Question: If yesterday morning at 6:00am the temperature was -4°C and this morning at the same time, the temperature was -10°C; assuming that temperature is a continuous and differentiable function, which of the following statements is a direct consequence of the Mean Value Theorem?

A) At some time between yesterday and this morning, the temperature was precisely -7°C.

B) The temperature remained between -20°C and 20°C throughout this time period.

C) The temperature took all the values between -4°C and -10°C but we cannot conclude that it took any value above -4°C or below -10°C.

D) At some time between yesterday and this morning, the temperature was decreasing at a rate of 0.25°C per hour.

E) The average rate of change of temperature is negative.

Logarithmic and Exponential functions

You can have a look at section 3.8 in the textbook. I don't really recommend reading that section in particular, though you're expected to know what's in box 3.17, 3.18, 3.19 and 3.20.

Example 2 and 4 are really childish, but good if you need basic practice with the above formulae.

Example 3 is already slightly more interesting, though notice how when they solve in part a) they end up saying it's roughly 40 years old when we could have already guessed that from the table. The thing that's really interesting by setting up this model is that we can actually be more precise and say it's close to 40.099 which then shows how really close to 40 years old it is (this is 40 years and 36 days to be precise).

Example 5 is good if you're a bit confused at how to manipulate exponentials and logarithms, though in that example, they only compute a horizontal tangent line, so conceptually, nothing new.

For now, we don't really care about logarithmic differentiation. It's a kind of nifty way to compute some derivatives and it will show up in the course eventually (not that it's hard now,it's just not that interesting). You could actually learn it by yourself.

Now, it terms of practice problems.

I like all the Review Questions (1-8) except for the last one.

The Basic Skills (9-22) are very similar to Webwork questions, you can practice that for sure and check your answers in Wolfram|Alpha.

Problems 23, 24, 25 are actually interesting, but easy applications. Those are baby versions of harder problems, you should be able to do those fairly easily. Spend time thinking how things relate to another instead of trying to find a formula that will just "do the job".

Problems 26 to 50 are Webwork like problems as well. Pick a topic you think you need to reinforce and practice.

Problems 51 to 58 are stupid and not interesting.

Problems 59 to 66 are again kind of Webwork like, some interesting skills to build as well.

Problems 67 to 74 are on logarithmic differentiation if you tried to study that already. Otherwise, come back later.

Problems 75 to 79 are much more interesting and actually constitute something worth spending time on that's more similar to homework questions, almsot worthy of being on an actual exam.

The Additional Exercices (80-87) aren't that interesting and are intended for people who love their graphing calculators mainly, so really not me.

If you have found other problems worth studying, post it in the discussion page so that I can have a look at them, I'll update this page if there's anything worth sharing with the rest of the class.

Question: If water lilies double every day and it takes them 60 days to cover the surface area of a lake, how much time was needed for them to cover half of the lake?

A) 20 days

B) 30 days:C) 31 days

D) 55 days

E) 59 days

Question: If the population at the end of the year is always 102% of the population at the beginning of the year, then which of the following is correct:

A) The population increases by 102% per year.

B) The yearly increase of population is 2%.

C) The instantaneous rate of change of the population is 2% of its size.

Practice problems and learning goals

Here are practice problems organized by learning goals. I'll keep adding them as we go. If you have any questions about the problems themselves, discuss them on the Math Forum. If you have questions about the choice of problems or suggestions for other problems, discuss that in the talk page of this page.

Unless stated otherwise, all the recommend problems are from the Briggs and Cochran - Calculus textbook.

Sometimes, some problems uses notions or concepts that we haven't studied in class yet. In that case, just skip and come back once you have the required tools (for example, we haven't talked about logarithms and exponentials, problems involving those should be studied once you're more comfortable with those functions).

Learning goals related to the theoretical content

• B1. the idea of limit and to evaluate limits involving basic functions using the limit laws, the squeeze theorem and/or the l’Hospital’s rule;
  • For the general concept of limits, section 2.2, exercices 1 to 14, 18 to 22.
• B2. the relationship between limits and asymptotes and to find asymptotes using limits;
  • For the relationship with vertical asymptotes, section 2.4, exercises 1 to 6, 15, 16, 33 to 36, 37 to 44.
  • To find vertical asymptotes using limits, section 2.4, exercises 17 to 26.
  • For the relationship with horizontal asymptotes, section 2.5, exercices 1 to 8, 37, 52, 53.
  • To find horizontal asymptotes using limits, section 2.5, exercices 9 to 36, 38 to 47.
• B3. the idea of continuity and to construct/determine functions that are continuous or discontinuous using the definition of continuity or theorems involving continuity;
  • For the conceptual part, the review exercices of section 2.6, 1 to 12, 55, 83, 84.
  • For verifying continuity of a given function, section 2.6, exercices 13 to 24, 85, 86, 88, 89.
• B4. the idea of derivative in terms of the slope of the tangent line to a curve, the rate of change of a quantity with respect to another quantity, and the limit definition of derivative;
  • For the concept of what is a derivative, section 3.1, exercices 1 to 10.
  • For relationships to the tangent line, section 3.1, exercices 11 to 32.
  • To compute derivatives using limits, section 3.1, exercices 49 to 52.
• B5. the idea of differentiability and to construct/determine functions that are differentiable or non-differentiable;
• B6. the graphical/numerical relationship between a function and its derivative;
• B7. to differentiate basic functions using the definition of derivative or the differentiation rules (derivative formulas, product rule, quotient rule, chain rule, logarithmic differentiation);
• B8. to implicitly differentiate an equation involving two variables and to find tangent lines to the graph of an implicit function;
• B9. to find the critical points and the local/absolute maxima/minima of a function defined on any open/closed interval;
• B10. to find the intervals of increase/decrease of a function using derivative tests;
• B11. to find the intervals of concavity, and to find the inflection points of a function defined on any interval using derivative tests;
• B12. to graph a function by analyzing the behaviour of the function using limits and derivatives;

Learning goals related to the applications of the course's content

• C1. the idea of linear approximation and to find linear approximations to functions;
• C2. the idea of approximation error and to estimate the error bound of a linear approxi- mation;
• C3. to solve application problems involving velocity and acceleration of moving objects, rate of change, economics, natural growth/decay, related rates, linear approximation and optimization by using basic mathematics, limits, and derivatives.
• C4. to properly read a theorem or an implication and construct the contrapositive statement;
• C5. to prove (or disprove) a statement or a mathematical formula by logical arguments without using the same statement or formula being proved;
• C6. to apply a theorem by satisfying its hypotheses and drawing logical conclusion, or by negating the conclusion and concluding that not all the hypotheses are satisfied;
• C7. the intermediate value theorem, Rolle’s theorem and mean value theorem, and to use these theorem to prove mathematical statements.