Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Math 115 Practice Final
Please workout each of the given problems. Credit will be based on the steps that you show
towards the final answer. Show your work.
Problem 1 Find y' for the following
A. y = x2 (1  x)5
Solution
We use the product rule first
y' =
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Name
MATH 115 MIDTERM III
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Without the use of your graphing calculator,
A) Determine the relative extrema if any
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
The Product and Quotient Rules
The Product Rule
Theorem:
Let f and g be differentiable functions. Then
(f(x)g(x)' = f(x)g'(x) + f '(x)g(x)
Proof:
We have
d/dx (fg)
f(x+h) g(x+h)  f(x) g(x)
= lim
Add and subtract f(x + h)g(x)
h
f(x+h) g(x+h)  f(x+h) g(x)
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
The Chain Rule
The Chain Rule
Our goal is to differentiate functions such as
y = (3x + 1)10
The Chain Rule
If
y = y(u)
is a function of u, and
u = u(x)
is a function of x then
dy
dy
du
du
dx
=
dx
In our example we have
y = u10
and
u = 3x + 1
so that
dy/dx
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
MATH 115 PRACTICE MIDTERM 1
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a
count
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Rates of Change
Velocity
Recall that the difference between speed and velocity is that velocity has direction and speed
does not. In other words, the speed is the absolute value of velocity. We have seen that the
secant line can be used to approximate the
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Related Rates
Related Rates (Definition and Process)
Another synonym for the word derivative is rate or rate of change. When you hear the
word rate you should identify d/dt, since rate always corresponds to the derivative with respect
to time.
To solve a
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
MATH 115 PRACTICE MIDTERM II
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Find
for the following
1  x2
A)
y =
1 + x2
Solution
We use the quotient rule we h
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Continuity
Continuity
If the limit exists at x = c, the function has some nice properties. However, even if the limit
exists, there may still be a hole. We define a function to be continuous at x = c is the limit
exists and the function agrees with the li
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
MATH 115
CALCULUS for SOCIAL and LIFE SCIENCE
Tuesday and Thursday 8:00 to 9:50 PM
Room D108
4 UNITS
Instructor Larry Green
Phone Numbers Office:
5414660 Extension 341
Internet
email:[email protected]
WWW: http:/www.ltcc.edu/depts/math/
Your Grades
Requi
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Limits
Example
Consider the function
x2  1
f(x) =
x2 +2x  3
If we plug in 1 we arrive at 0/0 which is undefined. What does this function look like near x =
1?
We can construct the following table:
x .9 .99 1.1 1.01 1.001
f(x) .487 .499 .512 .501 .5001
W
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Lines and Functions
Lines and Slope
Recall that the slope of a line through points (x1,y1) and (x2,y2) is the rise over the run or
y2  y1
m =
x2  x1
For a function f(x) the secant line between x = a and x = b is the line through the
points (a,f(a) and (
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Implicit Differentiation
Implicit and Explicit functions
An explicit function is an function expressed as y = f(x) such as
y = 2x3 + 5
y is defined implicitly if both x and y occur on the same side of the equation such as
x2 + y2 = 4
we can think of y as
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Higher Derivatives
The Second Derivative
The derivative of the derivative is called the second derivative.
There are two main ways of writing the second derivative. They are
d2y
f '(x)
and
dx2
The main benefit of the first notation is that it is easy to w
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Differentiation Rules
Constant Rule and Power Rule
We have seen the following derivatives:
1. If f(x) = c, then f '(x) = 0
2. If f(x) = x, then f '(x) = 1
3. If f(x) = x2, then f '(x) = 2x
4. If f(x) = x3, then f '(x) = 3x2
5. If f(x) = x4, then f '(x) =
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Definition of the Derivative
The Slope of a Secant Line
Example:
Consider the function
y = f(x) = x2
Then the secant line from x = 2 to x = 4 is defined by the the line that joins the two
points (2,f(2) and (4,f(4). This line has slope
f(4)  f(2)
16  4
Differential Calculus for Business and Life Sciences
MATH 115

Fall 2010
Graphing
The Distance Formula
If two points are in the plane, we can find the distance between them by using the Pythagorean
theorem. For example, if the points are (1,4) and (5,2), then the the figure below shows the right
triangle that helps us find the