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
