Math 202 – Review Sheet – Test 1
1. Understand and be able to work with the concept of a limit (page 450).
2. Understand and be able to work with limit properties (pages 452 – 454).
3. Recognize that the limit as x → a of a polynomial function is f (a).
4. Be able to evaluate the limit of a rational function as x → a when the value a causes the denominator to become 0.
 If possible, factor numerator and denominator and reduce the expression, canceling the problem factor, then
evaluate to find the limit.
 Alternatively, analyze the limit as x → a using example 1, pages 459 and 460, as a guide.
f ( x  h)  f  x 
for a given function.
h 0
h
5. Be able to evaluate lim
6. Understand and be able to evaluate a ‘one sided limit’ as x → a as the x approaches a either from the left or from
the right.
 Recognize that limits from the left and right must be the same if an overall limit exists for the function at
x = a.
 Recognize that a one sided limit will be the limit if a is an endpoint of the domain of the function.
7. Be able to evaluate a limit, if it exists, as a variable approaches positive or negative infinity.
 Understand and be able to work with shortcuts used to take the limit of a rational function or polynomial
function illustrated in examples on pages 462 – 464.
8. Be able to graph and find the limit (if it exists) of a case defined function at any point in the domain of the function.
1
x
9. Be able to work with e, that is lim (1  x) .
x 0
10. Understand and be able to work with criteria (page 467) that determine if a function is continuous at a given point.
 Note: A polynomial function defined over the real numbers is continuous at all points.
 Note: A rational function is continuous for all values of the variable that do not cause the denominator to
become 0.
o Understand the concept of infinite discontinuity (page 468).
11. Be able to apply the concept of continuity to the solution of inequalities as seen in the text in Sec. 1.4.
f ( x  h)  f  x 
at a specific value x = a is known as the derivative of the function f at x = a and
h 0
h
12. The lim
represents the slope of a line tangent to the graph of f(x) at the point (a, f(a)).
 Be able to find a slope of the tangent to a curve at a specific point by using this relationship.
 Be able to use this slope to write the equation of a line tangent to a curve at a specific point.
 Recognize and be able to use the various notation used to represent the derivative as summarized on page 484.
13. Be able to work with the shortcut rules for differentiation listed below:

Derivative of a constant, page 489,
d (c )
0
dx

Derivative of f(x) = xn, page 490,
d (x n )
 n( x ) n  1
dx

Constant factor rule, page 492,

Derivative of a sum or difference, page 493,

Product rule, page 507,
d [c  f ( x)]
 c  f ' ( x)
dx
d [ f ( x)  g ( x) ]
 f ' ( x)  g ' ( x)
dx
d [ f ( x)  g ( x) ]
 f ( x)  g ' ( x)  g ( x )  f ' ( x)
dx
d
   g ( x)  f ' ( x) 
f ( x)
g ( x)

Quotient rule, page 510,

Chain rule, page 516, If y = f (u) and u = g (x) ,

Power rule, page 517,
dx
g ( x)
f ( x)  g ' ( x )
2
dy
dy du


dx
du dx
d (u n )
du
 n  u n 1 
dx
dx
14. Understand and be able to work with practical applications that deal with the derivative as a rate of change.
Specifically, understand and be able to work with:




Function
Derivative
Distance
Cost
Revenue
Consumption
Velocity
Marginal Cost
Marginal Revenue
Marginal Propensity to Consume
Marginal Propensity to Save.
In addition, understand and be able to work with:

average cost as cost divided by quantity;

revenue as the product of price and quantity;

relative rate of change and % rate of change as ratios of the derivative of a function to the original function.