# Course Syllabus for Calculus

• CALCULUS COURSE SYLLABUS

I.                    Precalculus Topis--Informally

A.      The real line and order

B.      Absolute value and distance on the real line

D.      Factoring polynomials and finding zeros

E.       Fractions and rationalization

II.                  Calculus Topics

A.      Limits, their properties, and functions

1.       Given a function, the student will be able to calculate the domain and range.

2.       Given a function, the student will be able to recall the properties of limits and various techniques for evaluating the limits, to calculate the limit of the function when x approaches a certain value.

3.       Given a function, the student will be able to determine its continuity and one-sided limit.

B.      Differentiation

1.       Given a function, the student will be able to calculate the derivative of the function.

2.       Given a function that has an inverse function, the student will be able to determine the inverse of the function and its derivative.

3.       Given a problem involving velocity, the student will be able to solve the problem using derivatives.

4.       Given a composite function, the student will be able to determine its derivative by applying the chain rule.

5.       Given a product of functions, the student will be able to determine the derivative by applying the product rule.

6.       Given a function, the student will be able to determine the first derivative, the second derivative, the third derivative, etc.

7.       Given a quotient of two functions, the student will be able to determine its derivative by applying the quotient rule.

8.       Given trigonometric functions, the student will be able to determine its derivative.

9.       Given a polynomial, the student will be able to apply the rules to calculate the derivative of a polynomial.

10.    Given an equation, the student will be able to perform the implicit differentiation to calculate the derivative.

C.      Applications of Differentiation

1.       Given necessary information, the student will be able to set up equations and use differentiation to solve related rate problems.

2.       Given a non-continuous function, the student will be able to determine the points of discontinuity.

3.       Given a function, the student will be able to determine whether the function is increasing or decreasing on a given interval by using the sign of the first derivative.

4.       Given a function, the student will be able to determine the point of inflection using the second derivative.

5.       Given a function, the student will be able to sketch its graph on a graphing calculator, taking into consideration concepts such as domain and range, x-intercepts and y-intercepts, symmetry, points of discontinuity, vertical asymptotes, horizontal asymptotes, relative extrema, concavity, and points of inflection.

6.       Given a function, the student will be able to determine whether the graph is concave upward or downward by using the sign of the second derivative.

7.       Given a problem, the student will be able to apply Rolle’s Theorem and the Mean Value Theorem to solve it.

8.       Given a function, the student will be able to calculate its differential from the derivative and use differentials to obtain reasonable approximation to a problem.

9.       Given necessary information, the student will be able to use the maxima and minima theory to determine the maximum and minimum values.

D.      Integration

1.       Given the derivative of a function, the student will be able to use the notation for antiderivatives and will be able to determine the function.

2.       Given the derivative of a function, the student will be able to determine the function by using basic integration rules.

3.       The student will be able to calculate the area of a plane region by the integration method.

4.       The student will be able to apply the Fundamental Theorem of Calculus and the Mean Value Theorem for integrals to solve a given problem.

5.       The student will be able to apply the u- substitution, change of variables, and the general power rules for integration to solve a given problem.

E.       Applications of  the Definite Integral

1.       The student will be able to calculate the area of a region between two curves by the integration method.

2.       Given a three-dimensional solid, the student will be able to calculate the volume of a solid of revolution using the disc method or the washer method.

3.       Given a three-dimensional solid, the student will be able to calculate the volume of a solid of revolution using the shell method.

F.       Logarithm and Exponential Functions

1.       Given a logarithmic function, the student will be able to determine the derivative.

2.       Given a logarithmic function, the student will be able to determine the antiderivative.

3.       Given an exponential function, the student will be able to determine the derivative and the antiderivative.

G.      Techniques of Integration

1.       The student will be able to apply the basic rules of integration to solve a given problem.

2.       The student will be able to apply the rules of integration by parts to solve a given problem.

3.       Given a trigonometric function involving powers, the student will be able to determine the antiderivative.

4.       The student will be able to use trigonometric substitution to determine an antiderivative.