Concavity and Inection Points. Extreme Values and The
Second Derivative Test.
Consider the following two increasing functions. While they are both increasing, their concavity
The rst function is said to be concave up
behavior of a function near a point
The limit of a function f (x) at a point x = a describes the behavior of the function f (x)
near the point x = a.
If the values of f (x) accumulate near point
y = L when values of x approach
Antiderivative and The Indenite Integral
The idea of employing an inverse function is central when solving any equation:
x + 2 = 4 requires application of the function x 2, the inverse of x + 2, to the right hand side.
Thus x = 4 2 = 2
Section 7 Spring 2017
PROJECT 1 MA110 SECTION 7 BENDL
DUE MONDAY, MARCH 20, 2017
You may work with one other person in your section or by yourself. Please turn in
answers and all work on these sheets. Please staple all sheets together before handing
LIMITS GRAPHS . 2
1. Use the graph of f drawn below to answer the following
NOTE: OPEN DOTS AT (4,1.5) AND (1,1.5) AND (1,2).
' FILLED IN DOTS AT (4,1) AND (4,0).
(a) [im f(x)= ' (b) lim1f(x)=
(c) it foo-w - (d)
Write the given expression in logarithmic form.
1. 34mm 2. 4-23
3. (1.5)2=2.25 - 4. (1] =64
Write the given expression in exponential form.
5. log1,3(27) = 3 6. log2 8 m 3
- 1 . , 1
7. 10g16 2 m E 8. 10g3 [a] = 2
Rewrite the given expression as
FENCING # 1
YOU HAVE 1000 FEET OF FENCING TO FENCE THE RECTANGULAR REGION
SHOWN ABOVE. NOTE THAT THERE ARE TO BE TWO EXTRA FENCE LENGTHS
PARALLEL TO THE SOUTH SIDE, AS SHOWN. THE NORTH SIDE IS ADJACENT
TO A BUILDING AND DOES NOT NEED TO BE FENCED. THE
Jared is designing a new rectangular vineyard. He has 6420 feet of
fencing for his vineyard. He-plans to fence the entire rectangle and
put four extra lengths of fencing parallel to one side to make five
unequal subplots for different varieties of grapes
PRACTICE WITH FUNCTION DOMAIN
Find the domain of each of the following functions:
t t 6
1. g (t )
2. f (a)
3. h(t ) 5 t 3 2t
5. f (a)
7. f ( x ) x 3
a2 4a 4
a2 a 2
4. g (t ) 2t 5 1/2
6. g ( x )
3x 2 6x
x 2 6x 9
8. h( x )
GRAPH ANALYSIS PRACTICE
1. The function shown below models the pl-l level of the human mouth x minutes after
a person eats food containing sugar.
(a) Estimate the times (as x intervals) when the pH level is increasing and when
the pH level is decreasing
Rushabh D. Lagdiwala
a. Initial amount of medicine is 3000 mcg
Rushabh D. Lagdiwala ID#: 0621704
0.1 10.16568 1.016568
0.1 10.22558 1.022558
Karan Patel, ID # 588379
Hirvi Dalal, ID # 597367
Christopher Varghese, ID # 592957
Mathematical Analysis Lab , MA 102, Edward Reimers
Problem 1/Equation 1:
a. Initial amount of medicine is 3000 mcg
b. At t=2 hour
Implicit Dierentiation. Logarithmic Dierentiation
Implicit functions. If a function can be expressed as y = f (x) it is said to be in the explicit
form. However, in some cases, the variables x and y can be related with an equation F (x,
Innite Limits and Limits at Innity
Horizontal and vertical asymptotes
Example 1. Consider the function f (x) = x . This function is dened for every value of x except
x = 0. Although not dened at 0, it still may be relevant to know the
The Fundamental Theorem of Calculus.
The Total Change Theorem and the Area Under a Curve.
Example 2 of the previous section we approximated the distance traveled during some time
interval and given velocities at certain points of the in
Derivatives of Exponential, Logarithmic and Trigonometric
Derivative of the inverse function. If f (x) is a one-to-one function (i.e. the graph of f (x)
passes the horizontal line test), then f (x) has the inverse function f 1
Finding and Using Derivative
We have seen that the formula f (x) = limh0 f (x+h)f (x) is manageable for relatively simple
functions like a linear or quadratic. For more complex functions, nding the derivative using this
Continuous functions. Limits of non-rational functions.
Squeeze Theorem. Calculator issues.
Applications of limits
Continuous Functions. Recall that we referred to a function f (x) as a continuous function
at x = a if its graph has no h
Absolute Extrema and Constrained Optimization
Recall that a function f (x) is said to have a
relative maximum at x = c if f (c) f (x) for
all values of x in some open interval containing c.
However, that does not mean that the value f (
Higher Derivatives. Dierentiable Functions
The second derivative. The derivative itself can be considered as a function. The instantaneous
rate of change of this function is the second derivative. Thus, the second derivative evaluated a
The rate of change
Knowing and understanding the concept of derivative will enable you to answer the following
questions. Let us consider a quantity whose size is described by a function.
1. How fast is the quantity chang
Denite Integral The Left and Right Sums
In this section we turn to the question of nding the area between a given curve and x-axis on
an interval. At this time, this question seems unrelated to our consideration of indenite integrals in
Integrals of Exponential and Trigonometric Functions.
Integrals Producing Logarithmic Functions.
Integrals of exponential functions. Since the derivative of ex is ex , ex is an antiderivative of
ex . Thus
ex dx = ex + c
Recall that th
Areas between Curves
In this section we consider the area between two curves. Let f (x) and g(x) are two continuous
functions dened on the interval [a, b] such that f (x) g(x) for all x in [a, b]. If we consider a partition
of [a, b] wi
Increasing/Decreasing Test. Extreme Values and The First
Recall that a function f (x) is increasing on an interval if the increase in x-values implies an
increase in y-values for all x-values from that interval.
x1 > x2
More Rules - Product, Quotient and Chain
The Product Rule. Both calculus 1 and 2 courses would be much shorter if the derivative of
a product is a product of the derivatives. However, this is not true.
(f g) = f g
The dierential. Consider a function y = f (x) and the two points (x, f (x) and (x+h, f (x+h)
on its graph. Recall that x is sometimes used to denote the dierence h between the x-values and
y is used for the dierence