## Learning Goals

Below are the weekly learning objectives of MATH104/184.

### Weeks 12 & 13

Our main goal of this entire unit (Weeks 12 to 13), is to develop a deep understanding of linearization and more generally - Taylor polynomials. In weeks 11 and 12, you will cover material mainly from sections 4.4 and 8.7.

The specific learning goals are that by the end of the Week 12 and review homework, students should be able to:

1. explain linear approximation (also known as tangent line approximation and the linearization of a function);

2. use linear approximation or differentials to estimate the values of functions near a given x=a;

3. use linear approximation to approximate changes in the dependent variable given changes in the independent variable;

4. discuss the discrepancy with the linear approximation in terms of the second derivative (for example, whether it is an underestimate or overestimate of the exact value);

5. analyze the worst-case error for a linear approximation of a function using a formula based on the second derivative of the function; M(x-a)^2/2 where M is an upper bound on the absolute value of the second derivative on the interval defined by x and a.;

6. find the nth degree Taylor polynomial of a given function with a given centre x=a;

7. know the Maclaurin polynomials for e^x, sin x, and cos x;

8. use compostion to find the Maclaurin polynomials for e^{p(x)},sin (p(x)), and cos (p(x)), where p(x) is any polynomial.
9. use a Taylor polynomial to approximate the values of functions;

### Week 11

This week we will cover Optimization Problems (section 4.3 of our textbook). There will be some extra business related optimization problems posted as well. Some sections may be finishing up limits at infinity, infinity limits, and asymptotes from the Week 10 learning goals.

The specific learning goals for this week are that by the end of the week and webwork, students should be able to:

1. interpret the idea of optimization as the procedure used to make a system or a design as effective or functional as possible, and translate it into a mathematical procedure for finding the maximum/minimum of a function;

2. set up an optimization problem by identifying the objective function and all appropriate constraints;

3. use calculus to solve optimization problems, and explain how they used the constraints in the solution process;

4. justify that a soluiton is an extreme value over a domain that is not a closed interval; and

### Weeks 9-10

We cover the first and second derivative tests and curve sketching. This is material in sections 3.3 - 3.5, with some additional material from section 1.6 focused on asymptotes.

The specific learning goals for this week are that by the end of the week and review homework, students should be able to:

1. explain what an increasing or decreasing function is without terms from calculus [pg 136];

2. explain how the first derivative of a function determines where the function is increasing and decreasing and apply this to specific functions to determine their intervals of increase and decrease [Thm. 29];

3. use the first derivative test to identify local maxima and minima [Thm 30];

4. define concavity in terms of tangent lines.

5. explain how the second derivative of a function determines concavity and apply this to specific functions to determine where they are concave up and concave down, and to identify inflection points [Thm 31/32];

6. use the second derivative test to classify local maxima and minima [Thm 33];

7. identify any asymptotic behaviours a function may have: vertical asymptotes or horizontal asymptotes [Sec 1.6];

8. use calculus to sketch a graph of a given function [Sec 3.5].

### Week 8:

This week has two key topics. Price elasticity of demand and the Closed Interval Method for finding absolute maximimum and minimum values.

The specific learning goals for this week are that by the end of the week and review homework, students should be able to:

1. compute the price elasticity of demand and determine whether the demand curve is elastice or inelastic;

2. use the price elasticity of demand to determine the effect on revenue when there is a change in price;

3. maximize revenue by using price elasticity of demand;

4. define absolute maximum and absolute minimum and give examples of functions that illustrate these concepts;

5. state the Extreme Value Theorem, and give examples that illustrate their understanding of this theorem: (1) examples where the EVT applies, and (2) examples where the EVT does not apply, but functions have absolute maxima or minima;

6. define local maximum and local minimum and give examples of functions that illustrate these concepts;

7. define critical point and apply this definition to find and classify critical points of a given function;

8. find the absolute maximum and absolute minimum of a given continuous function on a closed interval.

### Week 7:

We now turn our focus to some incredible applications of Calculus. This week we will look at relative rates of change (i.e. growth rates as a percentage), velocity, acceleration, as well as average and marginal cost.

The specific learning goals for this week are that by the end of the week and review homework, students should be able to:

1. explain the difference between relative rates of change and absolute rate of change (or simply rate of change);

2. use the logarithmic derivative of a function f(x) to compute relative rates of change of f(x) per unit change of x;

3. work with velocity and acceleration as derivatives and solve problems involing them.

4. solve problems involving average and instantaneous growth rates.

5. explain the notion of marginal cost (or revenue or profit, etc.) in terms of the derivative of the cost (or revenue or profit) function.

6. explain the difference between marginal cost and the cost of producing one more unit.

7. solve problems involving average and marginal costs (or revenue and profit).

### Week 6:

This week we cover related rates (section 4.2) and inverse functions, specifically inverse trig functions (section 2.7).

The specific learning goals for this week are that by the end of the week and review homework, students should be able to:

1. set up and solve related rates problems. Given a draining tank, moving shadow, falling ladder, sailing ship problem, or provided a model (including equation) of another situation, students should be able to:

• identify all the variables involved, make appropriate choices when a variable takes on constant values, and describe how they relate (using equations if relevant and/or writing ashort paragraph);
• draw a diagram of the situation if needed;
• interpret rates in terms of derivatives with the appropriate variables; and
• derive an equation which describes how the relevant rates are related and differentiate that equation with respect to time in order to solve for the desired target rate.
2. find the derivative of arcsin x, arccos x, and arctan x by using implicit differentiation; and

3. find the derivative and any inverse function.

### Week 5:

We will cover the Chain Rule and implicit differentiation, sections 2.5 - 2.6. It is not necessary to cover the material on higher derivatives at this point (with regards to implicit differentiation). We also take a closer look at the derivatives of logarithmic and exponential functions. Lastly, we'll discuss logarithmic differentiation for dealing with functions whose base and exponent are both changing. The specific learning goals for this week are that by the end of the week and review homework, students should be able to:

1. state the Chain Rule, including its hypotheses, and identify when it can be used.

2. make use of the Chain Rule in computations.

3. explain what we mean by implicit differentiation and identify situations where they will use it.

4. carry out computations involving implicit differentiation.

5. find equations of tangent lines to graphs of implicitly defined functions.

6. find equations of normal lines to graphs of implicitly defined functions.

7. use the implicit differentiation to demonstrate the power rule for rational exponents.

8. work with the inverse properties of exp(x) and ln x;

9. use the derivatives of general logarithmic functions in computations;

10. use the derivatives of general exponential functions in computations;

11. use the technique of logarithmic differentiation;

### Week 4:

This week we look the derivative in more detail and learn some basic rules of differentiation. This is the material in sections 2.2 - 2.4 of our book.

The specific learning goals for this week are that by the end of the week and review homework, students should be able to:

1. state the (limit) definition of the derivative and use it to compute the derivative of a given function in simple cases (such as those like exercise 36 or 37).

2. sketch the graph of f ' given the graph of f.

3. explain using sketches of appropriate functions the relationship between continuity and differentiability.

4. determine when a function is not differentiable.

5. use the power, sum, and constant multiple rules to differentiate, for example, polynomials.

6. use the derivative of an exponential function.

7. know the definition of e as the base of the exponential function with the property its derivative is itself.

8. correctly state and use the product rule.

9. correctly state and use the quotient rule.

10. differentiate given functions using appropriate combinations of the rules of differentiation.

11. find equations of tangent lines to given functions at given points.

12. compute higher-order derivatives.

13. compute derivatives involving the basic trigonometric functions.

### Week 3:

This week we introduce continuity and the derivative. This is the material in section 1.5 and 2.1 of our book. Please note that we have skipped sections 1.6 (infinite limits) and will return to these section when we do curve sketching.

The specific learning goals for this week are that by the end of the week and homework, students should be able to:

1. explain what it means for a function to be continuous at a point. They should be able to correctly analyze whether a given function is continuous at a given point.

2. identify points of discontinuity for a given function.

3. know the way continuous functions behave under basic algebraic operations, and use these results to correctly identify whether or not a given function is continuous at a point. (See Theorem 8)

4. know the way continuous functions behave when they are composed.

5. identify whether or not a given function is continuous on a given interval. This includes identifying when a function is left- or right-continuous at the endpoints of a closed interval.

6. determine the value of a parameter which would glue two piecewise defined functions togheter nicely.

7. state the Intermediate Value Theorem and to apply it to simple situations such as determining whether or not a function has a zero in some interval.

8. explain the notion of instantaneous rate of change at a given point and its role as the slope of the tangent line at that point.

### Week 2:

Before formally introducing the derivative (week 3 or 4), we cover the material of Chapter 1 to gain an understanding of limits.

The specific learning goals for this week are that by the end of the week and review homework, you should be able to:

1. give an intuitive explanation of the process of taking a limit. You should be able to compute the limits of functions in simple cases, as presented in section 1.3 and 1.4.

2. compute the average rate of change of a function. You should be able to draw a diagram that illustrates this quantity.

3. draw a diagram to illustrate the process of computing an instantaneous rate of change of a function.

4. explain the relationship between finding average and instantaneous rates of change of a function and appropriate secant and tangent lines on graphs of this function.

#### Webwork Homework:

WeBWorK assignment 2 will open on Friday, September 16th at 6:00a and be due 10 days later on Monday, September 26th at 11:00p.

### Week 1:

In this introductory week, we will look at two important applications of mathematics in economics and finance. The first is an optimization problem involving revenue, costs, and profit. The second will be focus around (continuously) compounded interest.

The natural exponential and logarithm functions as these are extremely important in this course. Some discussion will be done in class, but students may need to review their high school notes.

The specific learning goals for this week are that by the end of the week and review homework, you should be able to:

1. explain revenue, costs, and profit for the case of a "simple business problem". You should be able to set up and solve a simple problem involving maximizing revenue and profit arising from a linear demand curve.

2. explain the difference between simple interest and compound interest.

3. explain the difference between nominal rate, effective rate, and continuous rate. You should be able to convert equivalent rates.

4. solve problems involving (continuously) compounded interest.

Students should also review their highschool notes and be able to:

• explain what an exponential function is. You should know the basic properties of exponential functions. You should be able to graph exponential functions. You should be able to solve basic equations involving exponential functions.

• explain what a one-to-one function is and how to test for this property graphically using the Horizontal Line Test.

• explain what an inverse function is. You should be able to determine the intervals on which a given function has an inverse (if they exist). Given the graph of function, you should be able to graph the inverse function, if it exists.

• describe the logarithmic functions as inverse functions of the exponential functions. You should know the basic properties of logarithmic functions that parallel those of the exponential functions. You should be able to graph logarithmic functions. You should be able to solve basic equations using logarithmic functions.

Note that the workshop next week will help to review this material.

#### WeBWorK Homework:

You will be asked to do your first WeBWorK assignment next week. It will be open on Friday, September 9th at 6:00a and be due 10 days later on Monday, September 19th at 11:00p.

Please note that there is an Introduction assignment this week to help you learn how to use the online homework system. It will open on September 5th and be due Saturday, September 10th at 11:00p