Project Problem:
Mat 210 Updated on February 5, 2012
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Review
Arithmetic
[pic] [pic]is true by definition. [pic], avoid the negative sign
[pic] is not a real number.
Indeterminate forms are: [pic], and [pic] is undefined, not indeterminate.
Algebra:
[pic] [pic] [pic] [pic] [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic]
n terms
Common denominator:
[pic]
Practice problems: Simplify using common denominator:
1. [pic] 2. [pic] 3. [pic] 4. [pic]
Function:
• Given [pic] then [pic], [pic], [pic].
• Given [pic] then [pic]
• Given [pic] then [pic]
• Given [pic] then [pic]
Section 10.1 Limits
In this section we will formulate some important rules for limits.
Following are examples where we use conjecture method:
Example 10.1 (1). Use conjecture to determine the limit [pic].
Solution. We use calculator to produce the table of values. Consider F(x)= [pic]
|x |-1 |-0.1 |-0.001 |-0.0001 |
|f(x) |3 |-5 |2 |1 |
Solution: The average value is [pic]
10.5 -10.6 Derivatives: Numerical, Graphical and Algebraic approach:
The slope [pic]of the secant PQ is defined by [pic]is also called the Newton quotient of the function f. Note that when h = 0, the fraction becomes 0/0, which is undefined. But choosing h = 0 corresponds to letting Q = P. When Q moves toward P along the graph of the function f, the x coordinate of Q, which is a+h, must tend to a, and so h tends to 0. Simultaneously, the secant line PQ tends to the tangent line to the graph at P.
Thus the definition [pic]
is called the slope of the tangent line at P to the graph of the function f. The number [pic]is also called the derivative of the function f at the point a.
The equation of the tangent line to the graph of [pic] at the point (a, f(a)) is [pic]
Example 1. Compute [pic]and [pic]of the given functions.
a. [pic] b. [pic] c. [pic]
Example 2. Find the slope of the tangent line to the graph of f at the specified points and also find the equation of the tangent line.
a. [pic] at (1, 1) b. [pic]at (0, 2) c. [pic]at (1, 1)
11.1 Derivatives of powers, sums and constant multiples:
The derivative of a function f is defined by the formula [pic], if this limit exists, we say that f is differentiable at x. The process of finding the derivative is called the differentiation.
Other symbols used in differentiation. The derivative of a function f is also represented by [pic]
Formula. The power rule. For the function of the type [pic], where a is a constant, then [pic].
Derivative of any constant is 0.
Example 1. Determine the derivative of the given power functions.
a. [pic] b. [pic] c. [pic]+5
Solution. a. [pic] b. [pic] c. [pic]
11.2 Marginal analysis:
For the following functions
C(x) = cost of producing x units
R(x) = px = revenue from selling x units, where p is the selling price per unit
P(x)= R(x) – C(x) = profit from producing and selling x units
Now we have the following functions
[pic]= Marginal cost at x
[pic]= Marginal revenue
[pic]= Marginal profit
11.3 Product and quotient rule:
Following are the differentiation rules for the functions with different operations.
[pic]
Example 1. Differentiate the functions (addition/subtraction).
a. [pic] b. [pic] c. [pic]
Solution. a. [pic] b. [pic] c. [pic]
Example 2. Differentiate the functions (multiplication/division).
a. [pic] b. [pic] c. [pic]
Solution. a. [pic] b. [pic] c. [pic]
11.4 The chain rule:
The chain rule is [pic]
Example 1. Find [pic]by chain rule.
a. [pic] b. [pic] c. [pic]
Solution. a. [pic]
b. [pic]
c. [pic]
11.5 Derivatives of logarithmic and exponential functions
Formula. The exponential rule. For the function of the type [pic], where a is a constant, then [pic]. Derivative of [pic] is [pic] .
Example 1. Find [pic].
a. [pic] b. [pic]
Solution. a. [pic]
Formula. For the logarithmic function[pic],[pic].
Derivative of [pic] is [pic] .
Example 2. Find [pic].
a. [pic] b. [pic]
Solution. a. [pic]
11.6 Implicit differentiation:
Functions like [pic], where x can not be written as independent variable with respect to another variable y. In this case direct differentiation is either difficult or impossible. To find derivative we will use implicit differentiation using chain rule. Following examples will illustrate the situation.
Example 1. Find [pic].
a. [pic] b. [pic]
Solution. a. [pic]
b. [pic]
Example 2. Find the slope of the tangent line to the curve [pic] at (1, 3). Also compute the second derivative.
Solution. [pic]
For the second derivative we consider again [pic]
Taking derivative we find [pic]
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