Expectations of students in science and engineering
The following list of abilities reflects the basic precalculus skills important for success in our calculus sequence. In addition to these is the ability to read carefully and interpret what has been read in the context of solving a problem. The student should be able to demonstrate these skills without a calculator.
The list includes links to reviews of many of the key topics.
In many problems done in a calculus course, often more of the actual work has its basis in precalculus rather than in calculus. Knowing this material before starting calculus makes its mastery much easier.
- Perform algebraic operations using the correct order of operations
Here is the standard phrase to recall the sequence of operations:
Please Excuse My Dear Aunt Sally
- Perform the operations inside a pair of parentheses first
- Then exponents
- Then multiplication and division, from left to right
- Then addition and subtraction, from left to right
- Be able to graph quadratic and cubic functions, ellipses, circles, and hyperbolas.
- Be able to manipulate algebraic expressions including using rules of exponents.
- Be able to complete the square of a quadratic expression and recognize when completion of the square is appropriate.
- Be able to determine the domain and range of a function.
- Understand the function concept including the composition of functions and be able to recognize the functions from which a given function is composed.
- Be able to determine the intersection of two lines or a line and a quadratic function.
- Be able to determine the equation of a line and understand when lines are parallel or perpendicular in terms of their slopes. Master the use of the point slope form of the equation of a line.
- Be able to solve linear inequalities and quadratic equations including equations which arise in novel circumstances.
- Be able to use properties of logarithmic and exponential functions to simplify expressions and solve equations. The inability of students to work with these functions is a problem among many entering students.
- Be familiar with the graphs of logarithmic and exponential functions, and the inverse relationship between logarithmic and exponential functions.
- Know the definitions of the trigonometric functions, be familiar with their graphs and periodicity, be able to evaluate trigonometric functions using standard triangles, and know basic trigonometric identities including the Law of Cosines and the double angle identities. Radian measure will be used almost all of the time when dealing with trigonometric functions. The inability to evaluate trigonometric functions and use standard identities is one of the greatest deficiencies among entering students.
- Know the Pythagorean Theorem and be able to apply it.
- Be able to recognize and use proportional relationships including those derived from similar triangles.
- Be able to use knowledge of basic plane and solid geometric figures to express relationships among them, e.g. when one is inscribed in another.
- Be able to form the converse and the contrapositive of a proposition and to recognize that the original proposition and the contrapositive are logically equivalent, but that the original proposition and the converse are logically unrelated.