MTH 175 Syllabus
Division: Arts and Sciences Date: February 2014
Curricula in Which Course is Taught: Science, Liberal Arts
Course Number and Title: MTH 175, Calculus of One Variable I
Credit Hours: 3 Hours/Wk Lecture: 3 Hours/Wk Lab: Lec/Lab Comb: 3
I. Catalog Description: Presents differential calculus of one variable including the theory of limits, derivatives, differentials, antiderivatives and applications to algebraic and transcendental functions. Designed for mathematical, physical, and engineering science programs.
II. Relationship of the course to curricula objectives in which it is taught: This course allows a student to use logical and mathematical reasoning in problem solving in theoretical and applied areas that require critical thinking.
III. Required background: A placement recommendation for MTH 175 and four units of high school mathematics including Algebra I, Algebra II, Geometry and Trigonometry or equivalent. Non-developmental through MTE 9.
IV. Course Content:
B. Linear Models and Rates of Change
C. Functions and Their Graphs
D. Fitting Models to Data
E. Inverse Functions
F. Exponential and Logarithmic Functions
G. A Preview of Calculus & Tangent line problem
H. Finding Limits Graphically and Numerically
I. Evaluating Limits Analytically
J. Continuity and One-Sided Limits
K. Infinite Limits
L. The Derivative and the Tangent Line Problem
M. Basic Differentiation Rules and Rates of Change
N. The Product and Quotient Rules and Higher-Order Derivatives
O. The Chain Rule
P. Implicit Differentiation
Q. Derivatives of Inverse Functions
R. Related Rates
S. Newton’s Method
T. Extrema on an Interval
U. Rolle’s Theorem and the Mean Value Theorem
V. Increasing and Decreasing Functions and the First Derivative Test
W. Concavity and the Second Derivative Test
X. Limits at Infinity
Y. A Summary of Curve Sketching
Z. Optimization Problems
V. Learner Outcomes
Upon completion of the course the students
will be able to:
1. Demonstrate the ability to sketch and recognize basic algebraic and transcendental functions.
2. Sketch the graph of one or more equations and state graphical information displayed.
3. Write equations of lines given particular conditions.
4. Interpret slope as a ratio or as a rate in a real-life application.
5. Use function notation to represent and evaluate a function stating its domain and range using proper interval notation and apply to real life situations.
6. Identify different types of transformations of functions.
7. Classify functions, recognize combinations of functions, and evaluate composite functions.
8. Interpret mathematical models for real-life data and fit a linear, quadratic, or trigonometric model to real-life data.
9. Demonstrate the ability to solve higher level equations and absolute value equations.
10. Demonstrate an understanding of what calculus is about, how it compares to precalculus, and how both the tangent line and area problem are basic to calculus.
11. Estimate a limit using a numerical or graphical approach.
12. Know different ways a limit can fail to exist.
13. Know and use the formal definition of a limit; complete a _-_ proof.
14. Evaluate a limit analytically by applying the properties of limits, dividing out and rationalizing techniques, and the Squeeze Theorem.
15. Develop and use a strategy for finding limits.
16. Demonstrate an understanding of continuity at a point, on an open interval, or on a closed interval, and apply the properties of continuity.
17. Determine and evaluate one-sided limits.
18. Understand and use the Intermediate Value Theorem.
19. Determine infinite limits from the left and from the right; apply infinite limit properties.
20. Find and sketch the vertical asymptotes of the graph of a function.
21. Find the slope of the tangent line to a curve at a point.
22. Use the limit definition to find the derivative of a function.
23. Understand the relationship between differentiability and continuity.
24. Apply the Constant Rule, Power Rule, the Constant Multiple Rule, and/or the Sum and Difference Rules to find the derivative of a function.
25. Find the derivatives of the sine and cosine functions.
26. Use derivatives to find rates of change. Understand the velocity and position functions.
27. Apply the Product Rule and Quotient Rule to find the derivative of a function.
28. Find higher-order derivatives of a function.
29. Understand and apply the Chain Rule to find the derivative of a composite and trigonometric function.
30. Find the derivative of a function using the General Power Rule.
31. Simplify the derivative of a function using algebra.
32. Ability to distinguish between functions written in implicit and explicit form.
33. Use implicit differentiation to find the derivative of a function.
34. Find related rates and use related rates to solve real-life problems.
35. Understand the definition of extrema of a function on an interval and relative extrema on an open interval.
36. Find the extrema on a closed interval.
37. Understand and use both Rolle’s Theorem and the Mean Value Theorem.
38. Determine intervals on which a function is increasing or decreasing.
39. Apply the First Derivative Test to find relative extrema of a function.
40. Determine intervals on which a function is concave upward or concave downward.
41. Find any points of inflection of the graph of a function.
42. Apply the Second Derivative Test to find relative extrema of a function.
43. Determine finite and infinite limits at infinity and state the horizontal asymptotes, if any, of the graph of a function.
44. Analyze and sketch the graph of a function without the aid of a calculator, stating all pertinent information about the sketch.
45. Demonstrate the ability to solve applied minimum and maximum problems (optimization).
46. Approximate a zero of a function using Newton’s Method.
47. Understand the concept of a tangent line approximation.
48. Compare the value of the differential, dy, with the actual change in y, _y.
49. Estimate the propagated error using a differential.
50. Find the differential of a function using differentiation formulas.
51. Find the inverse of a function; derivative of an inverse function.
52. Demonstrate properties of logarithmic and exponential functions; find their derivatives.
Test, quizzes, written assignments
VII. This course supports the following objectives:
DCC Educational Objectives: