MATH 250 - Calculus and Analytic Geometry I
4 Credit: (4 lecture, 0 lab, 0 clinical) 4 Contact Hours:
This is the first course in a two-semester calculus sequence. Topics included are limits, continuity, differentiation, differentiability, optimization, related rates and modeling. In addition, definite integrals and theorems involving definite integrals will be introduced. This course stresses conceptual understanding and multiple ways of representing mathematical ideas.
OFFERED: fall semesters
Course Goals/ Objectives/ Competencies:
Goal 1: Examine limits and continuity.
- paraphrase the meaning of a limit.
- describe the limit of a function using correct notation.
- determine the limit of a function and identify when the limit does not exist both graphically and numerically.
- explain the relationship between one-sided and two-sided limits.
- describe limits at infinity and infinite limits using correct notation.
- evaluate the limit of a function using properties of limits and algebraic techniques.
- evaluate the limit of a function using the squeeze theorem.
- explain the three conditions for continuity at a point and write the definition of continuity at a point.
- give an example of the intermediate value theorem.
- describe the epsilon-delta definition of a limit.
- apply the epsilon-delta definition to find the limit of a function.
- describe the epsilon-delta definitions of one-sided limits, infinite limits, and limits at infinity.
Goal 2: Investigate the concept of a derivative.
- explain the difference between average velocity and instantaneous velocity both numerically and graphically.
- apply a difference quotient to calculate the average rate of change of a function on an interval and interpret this numerically and graphically.
- identify the derivative at a point as the limit of a difference quotient.
- calculate the derivative of a function at a point analytically, estimate it numerically, and interpret the results.
- calculate and interpret the derivative in applications using appropriate units.
- define a derivative function.
- distinguish between the derivative of a function and the derivative of a function at a point.
- graph the derivative function from the graph of a function.
- explain and interpret higher-order derivatives.
- describe the connection between derivatives and continuity.
- describe the characteristics a function must exhibit to be non-differentiable at a point and verify this using limits.
Goal 3: Identify differentiation rules.
- apply the constant, constant multiple, power, sum, difference, product, quotient, and chain rules to find the derivative of functions where appropriate. These include constant, linear power, exponential, trigonometric, and logarithmic functions.
- use the limit definition of the derivative function to prove the basic differentiation rules.
- use logarithmic differentiation to determine the derivative of complex functions.
- calculate the derivative of an inverse function and determine the derivative of inverse trigonometric functions.
- apply implicit differentiation.
Goal 4: Employ derivatives to solve applications.
- model and solve related rates problems.
- describe the linear approximation to a function at a point and model the approximation.
- define and calculate differentials and draw a graph that illustrates the use of differentials to approximate the change in a quantity.
- distinguish local extrema from global extrema and use critical points to locate local and absolute extrema over a closed interval.
- apply the first and second derivative tests.
- explain the relationship between a function and its first and second derivatives.
- determine all points of inflection of a function and identify intervals of concavity.
- describe the significance of the Mean Value Theorem and state three important consequences of the theorem.
- model and solve optimization problems.
- apply L’Hopital’s rule to determine limits.
Goal 5: Distinguish between definite and indefinite integrals.
- explain the terms and notation used for an indefinite integral and find the general antiderivative of common functions.
- use antidifferentiation to solve simple initial-value problems.
- apply Riemann sums to approximate area and use sigma (summation) notation to write and calculate these sums.
- state the definition of the definite integral and describe the terms associated with it.
- use geometry and the properties of definite integrals to evaluate them.
- describe the relationship between the definite integral and net area.
- calculate the average value of a function.
- state the meaning of and apply the Fundamental Theorem of Calculus, Part 1.
- state the meaning of and apply the Fundamental Theorem of Calculus, Part 2.
- explain the relationship between differentiation and integration.
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