General Information:
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- Instructor(s): Paul Tsopméné
- Instructor(s) Email: paul.tsopmene@ubc.ca
- Instructor(s) Office: FIP 324
- Duration: Term 1 Winter 2020 (Sep-Dec 2020)
- Delivery Modality: On Zoom. The link is posted on Canvas.
- Classroom Schedule: Mon, Wed, Thu: 5 PM – 6 PM.
- Office Hours: Mon, Wed, Thu: 2:30 – 4:30 PM, Tue: 4 – 6 PM. These will be held on Zoom (The link is posted on Canvas). I am also available by email or appointment if these times do not work for you.
- Course Website: Course materials are available on Canvas. My primary method of communication is through Canvas messages/emails. Make sure you check this website regularly.
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Course Objective: To enable the student to:
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- Understand the fundamental concepts of differential calculus and applications to marginal analysis and elasticity of demand.
- Learn the most important techniques for calculating derivatives.
- Practice how to apply these techniques to model and solve various problems.
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Course Overview: This course will focus on single-variable differential calculus and its applications. While the list of topics aligns well with a general calculus course, most of the concepts will be presented through applications in Management and Economics. The course will start off with linear functions. In this chapter, we will learn how to find the slope between two points and the equation of a line. We will also learn how to find the equilibrium quantity, the equilibrium price, and the break-even quantity. In the second chapter, we will learn how to sketch the graph of a quadratic function and find the maximum profit/revenue, solve exponential and logarithmic equations, and solve (continuous) compound amount/interest problems. Calculus really starts in the third chapter with the concept of limits. We will learn how to find limits using rules and show that a function is continuous/discontinuous using limits. Then we will define the core concept of this course: the derivative. And we will learn how to find it using the limit definition, interpret the derivative geometrically (as the slope of the tangent line), interpret the derivative as the instantaneous rate of change, and sketch the graph of the derivative of a function defined by a graph. In the fourth chapter, we will learn how to find derivatives using rules (constant rule, power rule, constant multiple rule, sum/difference rule, product rule, quotient rule, and chain rule). We will also learn how to find and interpret the marginal cost, revenue, profit, average cost, average revenue, and average profit by using the concept of derivatives. In the fifth chapter, we will cover some applications of the derivative. We will learn how to find the intervals where a function is increasing/decreasing, find the relative extrema, find the intervals where a function is concave upward/downward, find the point of diminishing returns, and sketch the graph of a function. We will also learn how to maximize the profit/revenue function and minimize the cost function. The last chapter will cover more applications of the derivative. We will learn how to find the absolute maximum and minimum of a function on a closed interval, solve optimization problems, use implicit differentiation to solve related rate problems, find and interpret the elasticity of demand, use differentials to approximate the change in cost, revenue, and profit, and use Newton’s method to approximate solutions of algebraic equations.
Content: Topics include
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- Linear Functions: Slopes, Equations of Lines and Applications.
- Nonlinear Functions: Quadratic, Exponential and Logarithmic Functions.
- The Derivative: Limits, Continuity, Rates of Change, Definition of the Derivative, and Graphical Differentiation.
- Calculating the Derivative: Techniques for Finding Derivatives, Product, Quotient, and Chain Rules Derivatives of Exponential and Logarithmic Functions.
- Graphs and the Derivative: Increasing and Decreasing Functions, Relative Extrema, Concavity, the Second Derivative Test, and Curve Sketching.
- Applications of the Derivative: Absolute Extrema, Optimization, Implicit Differentiation, Related Rates, Elasticity of Demand, Differentials, and Newton’s Method.
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Learning Outcomes: Upon successful completion of this course, students will be able to:
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- Understand the basic idea of differential calculus: finding the slope of a tangent line to a curve.
- Evaluate limits both algebraically and geometrically. In particular, students should be able to find “limits of the form 0/0” and limits at infinity.
- Determine whether a function is continuous or not. And find the points of discontinuity (if any).
- Find derivatives by using the limit definition.
- Find derivatives graphically.
- Find derivatives by using rules (constant rule, power rule, constant multiple rule, sum/difference rule, product rule, quotient rule, chain rule).
- Find the slope of a tangent line to a curve.
- Find the equation of the tangent line to a curve at a given point.
- Find the (instantaneous) rate of change of a quantity.
- Find and interpret the marginal X, where X could be the cost, revenue, profit, average cost, average revenue, or average profit.
- Find the intervals where a function is increasing and where it is decreasing.
- Find the local or relative extrema.
- Find the maximum profit/revenue and the minimum cost.
- Find the intervals where a function us concave upward and where it is concave downward.
- Find the point of diminishing returns.
- Sketch the graph of a function.
- Find the absolute maximum and minimum of a function on a closed interval.
- Solve optimization problems, that is, problems where it is asked to maximize or minimize a quantity.
- Find the derivative of implicit functions by using the implicit differentiation method.
- Solve related rates problems. That is, problems where there two quantities are involved, and the rate of change of one quantity is given, and it is asked to find the rate of change of the other quantity.
- Calculate and interpret the elasticity of demand.
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Learning Resources:
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- Textbook (Optional): Calculus with Applications, by Margaret L. Lial, Raymond N. Greenwell, and Nathan P. Ritchey, 11th edition, Pearson. This is optional, as the lecture contents and the practice problems I will post on Canvas will be enough.
- Practice Problems: Practice problems and detailed solutions as well as reviews of the relevant theories will be posted on Canvas every week. While completion of these problems is essential for the success in this course, you do not hand them in for grading.
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Course Outline: More information can be found in the course outline and the document titled “online test information’’.