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.
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