Things I think you know and can do for Math 157
Lines.
I assume you can find the slope of a line through two distinct points, what it
means, what it tells you, when it exists and when it doesn’t. I assume you know the
relationship between the slopes of parallel and perpendicular lines.
I assume you are familiar with the three standard equation forms for straight lines
i.e. point – slope, slope – intercept, and standard. I don’t care which form you use but you
must be able to work with any of these forms. I prefer point – slope but I will accept any
(correct) answer in any form (and then the disclaimer) unless the problem asks
specifically for a specified form.
Algebra.
You must be able to factor quadratics with integer roots (zeros). You must be able
to factor higher degree polynomials which reduce to factoring a quadratic with integer
roots. For example:
x 4  5x 3  6x 2  x 2 (x 2  5x  6)  x 2 (x  2)(x  3)
You must be aware of the subtleties of “=” particularly with rational expressions.
x(x  1)
For example, why x 
is not true.
x 1
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A great deal of what goes on in the calculus involves inequalities consequently
you should be proficient with them. For example:
1 
Solve  x . If you get x  1 or x  1 then you have some work to do.
x
Solve x 2  x  6
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Of particular
concern
will be expressions involving absolute values. As a
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 consequence, it would be nice (read mandatory) to have a working definition of the
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absolute value. For example:
1
x
Solve
x
Rewrite without using the absolute value bars: x 2 1
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Another related problem is that of the words “and” and “or” in mathematics. Is
the solution to x 2  x  6  0 “ x  2 and x  3 ” or “ x  2 or x  3 ” and of course,
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why?
Trigonometry.
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You must be comfortable with radian measure and should be familiar with the
    2
standard angles in radian measure (e.g. , , , , ,etc.)
6 4 3 2 3

You must know the definitions of the six trig functions and be able to use them (I
don’t care if you think in terms of right triangles, arbitrary circles, or the unit circle but
you need to be able to go from one to another, much like the forms of straight lines).
You must know (or be able to quickly determine) the exact values of each trig
function at each of the standard angles. Personally, I know the sine and cosine and can
quickly compute the rest if called upon to do so.
You must know (or be able to quickly derive)
cos2 (x)  sin 2 (x) 1
cos(x  y)  cos(x)cos(y)  sin( x)sin( y)
sin( x  y)  cos(x)sin( y)  cos(y)sin( x)
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Now, taking advantage of the fact that cosine is an even function and sine is and
 odd function (so I assume you know what that means) for the third and fourth identity
 below and some tinkering you should be able to deduce the following which can be
handy on occasions.
sin( 2x)  2cos(x)sin( x)
cos(2x)  cos2 (x)  sin 2 (x)
cos(x  y)  cos(x)cos(y)  sin( x)sin( y)
sin( x  y)  cos(y)sin( x)  cos(x)sin( y)
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1 tan 2 (x)  sec 2 (x)
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