Math 142 - Exam 3 Concepts and Review (3.4-4.4)

power antiderivative rule ∫xⁿdx=

(1/n+1)xⁿ⁺¹ +C

what is the formula to determine ∆x (∆t)?

(b-a)/n ; n : number of subintervals (rectangles)

Steps to first derivative test

1. determine domain of f 2. find partition numbers 3. determine which partition numbers are in domain --- these are the critical values 4. create a sign chart of f' in domain of f

3 steps of closed interval method

1. determine domain of f and check that the function is continuous on the closed interval. if f is not continuous on the interval we cannot use this method 2. find the critical values of f. recall that these are the x values where f'(x) =0 or f'(x) DNE, and they're in the domain of f 3. evaluate f at the critical values in the interval and the endpoints of the interval, and determine the largest (absolute max) and smallest (abs min) values.

steps of finding a specific antiderivative

1. find the most general antiderivative 2. solve for the constant of integration C using the point given 3. substitute this value of C back into general antiderivative

what is the area of a semicircle

A=½πr²

what is the equation for revenue

Revenue = Price x Quantity

surface area of a box w open lid

S= x^2 +4xy

local or absolute extrema?: function can only have at most one of this extrema (y value)

absolute extrema

local or absolute extrema?: occur inside the function at max/min or at an end point

absolute extrema

what is constant C called

constant of integration

this equation is what you know

constraint equation

function has absolute minimum if its values go from

decrease to increase

du and dx are called

differentials

f(c) is an absolute minimum of f if...

f(c)≤ f(x) for all x in the domain of f

f(c) is an absolute maximum of f if ....

f(c)≥f(x) for all x in the domain of f

t or f : local and absolute extrema can occur at a hole

false

t or f: when problem says to find the absolute extrema of f it means the answer will be the x value of the point

false; the answer will be the y value of the max/min

t or f: the x*i value is the width of the rectangle

false; the x*i value is the height of the rectangle and the ∆x value is the width

t or f: in order to use case 2 there can be more than one exponential chain rule in the integrand

false; there can only be one b^g(x)/e^g(x)

t or f: when using u substitution with definite integrals the values of integrals given are okay to use

false; they are in terms of x so you would have to change them into terms of u by plugging the x values into the equation that equals u

fourth step of optimization process:

find the critical values of the objective function then move on to either use CIM (buis) or FDT/SDT (geometry)

third step of optimization process

find the interval on which the objective function must be optimized. to do this consider the context of the problem and restrict the variables to determine a realistic domain for the objective function

first step of solving optimization problem: geometry:

identify the quantity you want to max/min and the quantity that you know; draw a picture with labels and introduce variables; determine the relationships between the variables to create the objective function and possibly a constraint equation

first step of solving optimization problem : buisiness

identify the quantity you want to max/min and the quantity that you know; introduce variables; determine the relationships between the variables to create the objective function and possibly a constraint equation

extreme value theorem

if a function is continuous on a closed interval [a,b], then it must have both an absolute maximum and absolute minimum on the interval

first part of fundamental theorem of calculus

if f is continuous on an interval [a,b] and the function F is defined by F(x)= ∫a^x f(t) dt - where a≤x≤b then F'(x) = f(x) on the interval [a,b]

part two of the fundamental theorem of calculus

if f is continuous on the interval [a,b] and F is any antiderivative of f then ∫a^b f (x) dx = F(b) = F(a)

function has absolute maximum if its values go from

increase to decrease

what does dx indicate

integration w respect to x

f(c) is an absolute extremum of f if f(c)...

is an absolute max or absolute min

when is the area of regions between the function and the x axis considered to be a negative value

it is a negative value where the function is negative

constant antiderivative rule ∫kdx =

kx+C

local or absolute extrema? : occur inside the function

local extrema

local or absolute extrema?: multiple of this extrema can occur in a function (y value)

local extrema

this function is what you want

objective function

what is the function we are maximizing or minimizing called

objective function

when can Reimann sums be used to estimate the net area ?

when the function values are negative

"that" means to find the

x value

"where" means

x value

the width of the subintervals (rectangle) is referred to as

∆t (in a velocity time graph) but ∆x in general

integral sign

∫

case 3 of possible substitution cases

∫(1/g(x))×g'(x) dx

case 2 of possible substitution cases

∫b^(g(x))×g'(x) dx where b is any pos real number

special case of exponential antiderivative rule

∫e^xdx = e^x +C

case 1 of possible substitution cases

∫g(x)ⁿ×g'(x) dx where n is any real number w n≠1

special case of the power antiderivative rule

∫x⁻¹dx = ∫1/x dx = ln|x| +C

Properties of Definite Integrals

1. if the limits of integration are the same then there is no area ∫ⁿₙf(x) dx = 0 2. reversing the order of the limits of integration will change the sign of the definite integral ∫f(x) from [a,b] = −∫f(x) of [b,a] 3. the definite integral of a sum (or difference) is the sum (or difference) of the definite integrals 4. if we multiply a function by a constant and then find the definite integral it is equivalent to finding the definite integral of the function and then multiplying by the constant 5. a region can be divided into two separate regions and the area will remain the same whether we calculate it all as one region or as two separate regions ∫a^b = ∫a^c + ∫c^b

steps to perform substitution

1. let u = g(x) 2. write du = g'(x) dx 3. Rewrite the integral in terms of u and du 4. integrate 5. substitute back for u

exponential antiderivative rule ∫b^x dx=

1/ln(b)×b^x +C

the summation of all of the areas of "rectangles" is called a

reimann sum

second step of optimization process

state objective function in terms of one variable. if there is more than one variable then solve the constraint equation for one variable and substitute in the objective function

the process of reversing the chain rule to find an antiderivative

substitution or u-substitution

on a graph displaying the relationship between time and velocity how do you find distance?

the area under the line of this graph would equal to the distance

integrand

the function f that follows the integral sign

what is optimization

the process of finding an absolute max or min

integration

the process of finding the most general antiderivative of a function

what is a sample point

the x*i value we chose in each interval to determine the height of a particular rectangle

"is" means to find the

the y value

why do we use the closed interval method?

to compare the values of the function at the critical values in the interval and the endpoints of the interval

when do you use the closed interval method?

to find the absolute extrema of a function on a closed interval on which the function is continuous;

t or f : a function will have infinitely many antiderivatives

true

t or f : for geometric optimization problems FDT or SDT can be used in the fourth step

true

t or f : when you take the limit of the Reimann sum the answer will be exact and not an estimate

true

t or f: Reimann sum is used to find an ESTIMATE of the area between the x axis and curve/line of a function

true

t or f: every business optimization problem should have a closed interval so CIM will be used

true

t or f: if you increase the number of rectangles used in a Riemann sum the estimation will approach the exact answer

true

t or f: in the third step of optimization (buis problem) p is always greater than or equal to 0

true

t or f: it can be assumed that C= 0 because part 2 of the FTC says we can use any antiderivative to evaluate a definite integral

true

t or f: the area under a velocity curve is the change in position of an object during that intervals time

true

t or f: where f' is positive f increases and where f' is negative f decreases

true

t or f: where f'' is negative f' decreases and f is concave down and where f'' is positive f' increases and f is concave up

true

t or f: you can have local extrema even if f is not continuous

true

true or false: a function can only have one absolute max/min y value but can occur at multiple x value

true

t or f: local extrema cannot occur at an endpoint

true; function would need to occur on either side of the point even if a random point closed circle