What basic algebra skills and techniques are most important for calculus students to know?

Question

In my experience, algebra is one of the biggest stumbling blocks to calculus students. For instance, sign errors are common, and exponent laws (and log laws!) cause a lot of headaches.

Many courses have algebra pre-tests to let students know if they are "prepared" or not; however, these pre-tests often cover material that has nothing to do with the course itself.

Is there any published information on what algebra skills and techniques are most important for student success in calculus?

(I saw this question, but it covered a different aspect of the issue).

Answer 1

Sam Shah has this post on his blog: http://samjshah.com/2012/06/01/algebra-bootcamp-in-calculus/

I loved his list, but wanted to rewrite it a bit for myself. (Also, Sam finds it more effective to review the algebra ahead of time, while I think it's more effective to review once we see the need in our exploration of calculus.)

I teach my calculus course in an order that I think will help students learn. I have four units:

• Unit 1 includes history, graphing functions, slopes of tangent lines by approximation, algebraically finding the derivative using the limit (which we do not carefully define yet), seeing the similarities between velocity, rate of change, and slope, average versus instantaneous velocity, derivative from a graph, (estimated) derivative from a table of values.

• Unit 2 includes derivative properties needed for polynomials, graphing, limits, and trig derivatives.

• Unit 3 includes chain rule, derivatives of exponential functions, implicit differentiation, derivatives of inverse functions (ln x and inverse trig functions), and related rates.

• Unit 4 includes integration (finding area under the curve), anti-derivatives, fundamental theorem of calculus, and substitution method. If there is time we include volumes of rotation (which I think is a perfect ending for the course).

Algebra Skills needed for Unit 1

Algebra

• Determine the equation of a line given two points, or a point and a slope, or a graph of a line,

• Find the average rate of change over an interval given a function or its graph,

• Clearly express what is happening to an object given a position versus time graph,

• Evaluate f(x+h) for any given function f(x),

• Rationalize the numerator (to find the derivative of the square root function) ,

• Simplify complex fractions (to find the derivative of the 1/x function).

Algebra with Calculus Concepts

• Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or its graph,

• Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change,

• Sketch a velocity versus time graph given a position versus time graph,

• Construct the formal definition of the derivative by modifying the definition of slope,

• Apply the formal definition of the derivative to simple polynomials and to simple square root functions.

Algebra Skills needed for Unit 2

Algebra

• Multiply out the expression (x+h)n (necessary to understand the proof for the derivative of y=xn),

• Identify the holes, vertical asymptotes, x- and y-intercepts, horizontal or slant asymptote, and domain of any rational function,

• Sketch the basic shape of a rational function,

• Identify an equation for a rational function given a sketch of the function,

• Explain clearly what a hole and an asymptote are,

• Construct the equation of a piecewise function given its graph,

• Sketch the graph of a piecewise function given its equation,

• Work with inequalities,

• Give both triangle and circle definitions of sin x, cos x, and tan x, and explain how they’re related,

• Evaluate sin x, cos x, and tan x at all multiples of π/6 and π/4, without a calculator,

• Understand trigonometry identities, including and sin(x+h)=sin x cos h + sin h cos x,

• Graph y = sin x and y = cos x.

Algebra with Calculus Concepts

• Graph a polynomial or rational function, showing its maximums, minimums, and inflection points,

• Follow complicated logic (in the definition of limit).

Algebra Skills needed for Unit 3

Algebra

• Understand composition of functions,

• Use logarithm properties to “break apart” a single logarithmic expression into simple logarithms,

• Understand properties of exponents,

• Be able to graph exponential and logarithmic functions.

Algebra with Calculus Concepts

• Think in terms of composition of functions to determine outer and inner functions, in order to use the chain rule.

Algebra Skills needed for Unit 4

Algebra

• Work with summations.

Answer 2

Most problems I see (in classes after the first calculus courses) are basic misunderstandings in what is allowed to do with equations (and even worse with inequalities). There are also misconceptions on the meaning of strings of equations/inequalities, i.e., $p^2 < p^2 + u \le p^2 + p < p^2 + 2 p + 1 = (p + 1)^2$ (here $1 \le u \le p$). There are problems with the concept of a variable (i.e., it doesn't vary, it is a fixed, perhaps unknown value or some value we don't want to nail down for the sake of generality; note that this are two different notions).