We've had our first test in Calculus I. A few students had difficulty with the test. In many cases that was due to forgotten algebra and topics from precalculus. Use the following list to help you identify the topics that you may need to study.
Topics from algebra. We use algebra constantly. You've got to know
algebra well.
Simplify expressions and knowing when they can't be simplified. For example,
the square root of a sum is not the sum of the square roots, and a divided by b + c does not equal the sum of a/b and a/c.
Expanding products and powers. For example, (a + b)2
is equal to
a2 + 2ab + b2, not
a2 + b2
Factoring quadratic polynomials and other simple polynomials
Proper use of parentheses. When expressions aren't fully parenthesized,
multiplication and division have higher precedence than addition and subtraction.
For example, the product of sin x and 3x2 + 5
is not sin x 3x2 + 5. It's
(sin x)(3x2 + 5), and all the parentheses
are necessary unless you write the product in the other order,
(3x2 + 5) sin x
Techniques for solving inequalities and both equations and inequalities involving
absolute value. We often use
|a – x| < b to express symbolically that
the number x is within b of the number a.
The concept of function, functional notation and substitution. For example, you
should know what f(x +h) is when f(x) = 2x2 – cos x.
It's
2(x +h)2 – cos (x +h),
and all the parentheses are required.
Topics from trigonometry.
Understanding of trig functions of angles, especially sine,
cosine, tangent, and secant. Trig functions and the unit circle. Radians
Right triangles, trig functions sine, cosine, and tangent of
acute angles. Values of these trig functions for standard angles
of 0, π/6, π/4, π/3, π/2
Values of trig functions for angles other than acute angles
Basic trig identities. Pythagorean identities, trig functions in terms
of sines and cosines. Double angle formulas
for sine and cosine, addition formulas for sine and cosine
Exponential functions and logarithms.
Exponential functions. Growth of exponential functions
Laws for exponents. Manipulation of algebraic expressions involing
exponents, solving equations involving exponents
Logarithms and their relation to exponential functions
Laws for logs. Manipulation of algebraic expressions involing
logs, solving equations involving logs
Graphs of functions, limits, continuity, asymptotes
Reading limits from the graphs of functions, left limits and right limits, when
limits don't exist.
Reading continuity from the graphs of functions.
Recognizing limits from asymptotes of graphs, and graphing the asymptotes of
functions that have infinite limits and limits at infinity.
Limits and their formal definition
Evaluating limits, especially those of the form 0/0
The roles of epsilon and delta
How to find a value of delta that works for a given epsilon when the function is
a fairly simple function such as a linear function