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PRACTICE PROBLEMS: Differentiate the functions in problems 1 through 3.
4. Find the slope of the tangent line to the curve
at the points x =  2, 0, and 3 5. Find the values of x where the function
has a maximum or a minimum. ANSWERS:
SUMMARY The
following are the major topics covered in this chapter:
2.
Indeterminate forms:
Two
methods of evaluating indeterminate forms are (1) factoring and (2) division
of the numerator and denominator by powers of the variable. 3.
Limit theorems: Theorem
1. The limit of the sum of two functions is equal to the sum of the limits:
This
theorem can be extended to include any number of functions, such as
Theorem
2. The limit of a constant, c, times a function, fx), is equal to the constant, c, times the limit of the
function:
Theorem
3. The limit of the product of two functions is equal to the product of their
limits:
Theorem
4. The limit of the quotient of two functions is equal to the quotient of their
limits, provided the limit of the divisor is not equal to zero:
4.
Infinitesimals: A variable that approaches 0 as a limit is called an
infinitesimal:
The
difference between a variable and its limit is an infinitesimal: If
lim V = L, then lim V  L = 0 5.
Sum and product of infinitesimals: Theorem
1. The algebraic sum of any number of infinitesimals is an infinitesimal. Theorem
2. The product of any number of infinitesimals is an infinitesimal. Theorem
3. The product of a constant and an infinitesimal is an infinitesimal. 6.
Continuity: A function, f(x), is continuous at x = a if the following three
conditions are met:
7.
Discontinuity: If a function is not continuous at x = a, then it is said to be
discontinuous at x = a. 8.
Ways of expressing the derivative of y with respect to x:
9. Increment method for obtaining the derivative of y with
respect to x: 1. Set up the function of x as a function of (x +
x) and
expand this function. 2. Subtract the original function of x from the new
function of (x +
x). 3. Divide both sides of the equation by
x. 4. Take the limit of all the terms in the equation as
x approaches
zero. The resulting equation is the derivative of f(x) with
respect to x. General formula for the derivative of any expression in x:
11. Maximum or minimum points on a curve: Set the
derivative of the function, f(x), equal to zero and determine the values of the
independent variable that will make the derivative equal to zero. (Note: When
the derivative of a function is set equal to zero, that does not mean in all
cases the curve will have a maximum or minimum point.) ADDITIONAL
PRACTICE PROBLEMS Find
the limit of each of the following:
ANSWERS TO ADDITIONAL PRACTICE PROBLEMS
