MATH 250 - Calculus and Analytic Geometry I

4 Credit: (4 lecture, 0 lab, 0 clinical) 4 Contact Hours: [Math Level 7 ]

Objectives:  The student should be able to

1. describe a function numerically, graphically, algebraically and verbally.
2. state the properties of linear and exponential functions.
3. transform the graphs of functions with shifts, reflections and stretches and algebraically compose functions.
4. determine the inverse of a function algebraically, using a table of values and using graphical methods.
5. describe a logarithm as the inverse of the exponential function and solve a variety of problems involving logarithms.
6. state the properties of trigonometric functions and solve a variety of problems involving trigonometric functions.
7. summarize the graphical features of power functions.
8. summarize the characteristics of polynomial and rational functions.
9. determine the interval over which a function is continuous.
10. state the result of the Intermediate Value Theorem.

Goal 2:  Students will apply the concept of a limit and instantaneous velocity.

Objectives:  The student should be able to

1. determine the average velocity of an object on an interval and interpret the result both graphically and numerically using appropriate units.
2. compare and contrast average velocity and instantaneous velocity.
3. use the idea of instantaneous velocity to describe the meaning of finding a limit (or vice-versa).
4. describe what instantaneous velocity represents both graphically and numerically.
5. determine the limit of a function both graphically and algebraically.
6. describe when a limit does not exist.
7. state the formal definition of a limit.

Goal 3:  Students will apply the concept of a derivative.

Objectives:  The student should be able to

1. state the difference quotient and interpret this quantity as the average rate of change of a function on an interval and also interpret this quantity graphically.
2. apply the idea of a limit to find the instantaneous rate of change of a function at a point and interpret this number graphically.
3. define the derivative as the instantaneous rate of change.
4. state the limit definition of the derivative at a point.
5. determine the derivative of a function at a given point using the definition.
6. interpret the derivative of a function at a point both numerically and graphically.

Goal 4:  Students will demonstrate an understanding of the derivative as a function.

Objectives:  The student should be able to

1. state the limit definition of f ‘(x).
2. summarize what information the derivative of a function on an interval gives about the original function.
3. sketch the graph of the derivative given the graph of a function.
4. determine the derivative of constant, linear and power functions.
5. utilize the alternate notation of the derivative to make interpretations of the derivative using appropriate units.
6. describe what information the second derivative of a function on an interval gives about the original function.
7. interpret the second derivative as a rate of change.
8. state the formal definition of continuity of a function at a point.
9. state the relationship between continuity and differentiability.
10. describe what qualities a function which is non-differentiable at a point must exhibit.

Goal 5:  Students will determine derivatives of different types of functions.

Objectives:  The student should be able to

1. determine the derivative of a power function and an exponential function.
2. apply the product and quotient rules to find derivatives of certain functions.
3. determine the derivatives of trigonometric functions.
4. apply the chain rule where appropriate.
5. apply the chain rule to find derivatives of logarithms and inverse trigonometric functions.
6. determine derivatives using implicit differentiation.
7. use the equation of a tangent line to approximate a function near a point and estimate the error in the approximation.
8. apply L’Hopital’s rule to determine limits.

Goal 6:  Students will apply the first and second derivatives to analyze the behavior of families of functions and to solve optimization problems.

Objectives:  The student should be able to

1. apply the derivative to find critical points of a function.
2. determine if critical points are local extrema using the first derivative test.
3. determine if critical points are local extrema using the second derivative test.
4. determine all points of inflection of a function and identify intervals of concavity.
5. sketch graphs of functions based on information about its first and second derivatives.
6. determine what effect parameters have on different families of functions.
7. determine all local extrema and points of inflection on paramaterized functions.
8. apply tests for local extrema to find points to optimize a problem situation.
9. distinguish local extrema from global extrema.
10. apply mathematical modeling to determine a function to model a problem situation and use calculus to find the local or global extrema.
11. define and determine properties of hyperbolic functions.

Goal 7:  Students will demonstrate an understanding of the definite integral.

Objectives:  The student should be able to

1. describe the area under the velocity curve as displacement.
2. sketch on the graph of a velocity function the left and right hand approximation for displacement.
3. describe how to obtain total displacement of an object on an interval using limits.
4. determine the difference between the left and right hand approximation for displacement.
5. write left and right hand sums using Sigma notation.
6. define the definite integral as the limit of Riemann sums.
7. apply the definite integral to find the area under a curve and to find the average value of a function.
8. interpret the definite integral and give appropriate units associated with different applications.
9. explain the relationship between the derivative and definite integral as expressed in the Fundamental Theorem of Calculus.