Limits & Continuity Questions and Answers

3 Evaluate the following limits using limit laws Show each step a lim 2 2 b lim 1 2x35x 5x2 1 x 1 22x

Your demonstration will be to graph the following function x 0 x 0 f x 1 1 x 1 e

the properties of limits of functions to prove that x 7 lim x 3 x 3 and x 7 lim x 3 x 3

ete the equation below 11 cot cot O

the following statement of the function f x 3x

Solve exactly then gi 69x 1 7 T 2

You go to the doctor and he milligrams of dye remain in y radiation detector without s milligrams of the dye are in y were given the dye as soon a

X1 X2 Which expression corresponds to the shaded region A X1 UX2 B X1 UX2 C X1 UX2 OD X1 UX2 U

1 point In how many ways can 5 students be seated in a row of 5 chairs if Jack insists on sitting in the first chair Your answer is

A company produces two models of a surfboard a standard model and a competition model If the standard model is produced at a variable cost of 220 each and the competition model at a variable cost of 320 each and if the total fixed costs per month are 6 000 then the monthly cost function is given by C x y 6 000 220x 320y where x and y are the numbers of standard and competition models produced per month respectively Find C 24 12 C 40 4 and C 40 40 C 24 12 Simplify your answer

For the function f x y 9xy find f x h y f x y h MONTE CON f x h y f x y h

For the function f x y 8x 5y find f x h y f x y f x h y f x y h

Find the indicated value of the function 3 f x y z 4x 5y 5z 2 f 8 6 3 f 8 6 3

versions Within If 9 kg are placed on one side

Use the fractal like figure made using a recursive process with three steps

For what value of x is the following true log x 3 log x log 3 Your answer is X Question Help Video Message instructor

Consider the region R bounded by the parabola f x 2x x and the line y x a Find the exact area between the parabola and the line in R Show your steps and explain your reasoning b Set up an integral for the volume of the solid obtained by rotating R around the line x 2 You do not need to evaluate the integral Notes Your explanation should include i a rough sketch of the solid of revolution or region to consider ii Label your solid and show the radius es iii Clearly state the method you are using and why and iv Clearly show and justify your steps for setting up your integral Your final integral should be clearly shown

3 a Sketch the hyperbola and label the foci vertices and asymptotes 16 9 1

Question 2 Use your calculator to fin 2 In 5 In 3

f x 16 f x represents the position of an object for time x Find the average velocity over the the interval 4 2 Round your answer to four decimal places

et f x 3 3x 2 7x Find the equation of the secant line on the graph of f x between x1 3 and 2 11 Write your answer in mx b format

If A is the area of the intersection of the regions inside 2 cose and outside 1 then A is O a 4 3 3 6 Ob 3 3 Ox 1 3 d 6 16 A

Find the doubling time of the exponential quantity Q 400 1 1 You may leave your answer in exact form or round to 4 decimal places Be sure to show ALL the steps in your work

6 Use the change of base formula to change the following to a common logarithm Do not solve log 16 7 Solve for x Get x by itself 32x 1 5 8 Solve e x e 6 0 9 Solve for x log x log x 1 log 20 10 Find the inverse of f f x log x 1 ylemia bro noleg 21 gol 0

2 Find the limit Be sure to show work lim x ex 5

Math 125 v1 Test 4 Chapter 4 Score 100 You must show ALL work to receive full credit Problems are 4 points unless otherwise denoted 1 Find x log4 x 2 2 Evaluate the expression Simplify as much as possible log 6 log 15 log 20 3 Evaluate the expression log 16100 4 Use the laws of logarithms to expand the expression completely log 4 x y 6 Z 5 Use the laws of logarithms to combine the expression and simplify if possible log x 1 log x 1

The graph below shows a transformation of y 2 54 4 3 2 5 4 3 2 1 y 2 23 3 4 2 3 4 5 Write an equation for the graph above

Given the following function determine the equations of the vertical asymptotes of the principal cycle y 5 tan x

Animal populations are not capable of unrestricted growth because of limited habitat and food supplies Under such conditions the population follows a logistic growth model P t 1 ke ct where c d and k are positive constants For a certain fish population in a small pond d 1200 k 11 c 0 2 and t is measured in years The fish were introduced into the pond a How many fish were originally put in the pond 100 fish b Find the population after 10 20 and 30 years Round your answers to the nearest whole number 10 years 1057 X fish 20 years 1108 30 years 1197 c Evaluate P t for large values of t What value does the population approach as t P t 0 x Does the graph shown confirm your calculations P 1200 1000 800 600 X fish X fish 400 200

If a 60 year old buys a 1000 life insurance policy at a cost of 100 and has a probability of 0 976 of living to age 61 find th expectation of the policy until the person reaches 61 Round your answer to the nearest cent The expectation of the policy until the person reaches 61 is dollars

Complete the table of values rounded to two decimal places f x 1 7ex X 2 1 0 5 0 0 5 1 2 10000 Sketch a graph of the function 1 5 1 0 0 5 y 10 5 5 10 0 5 1 0 1 5 X 1 5 1 0 0 5 10 5 10 y 5 0 5 1 0 1

Find the sum of the follo 3 6 3 9 4 2 n

Let f C C be an entire function satisfying z 3 x 2y x 4 for all z x iy C Prove that f is a constant function

8 Find a parametrization for the curve The line through the points 1 2 ind 3 4 9 Use De Moivre s Theorem to find the

4 Prove the statement using definition of a limit lim x 5x 8 2 x 3

4 Prove the statement using s definition of a limit lim x 5x 8 2 any ly ngh ngay t i t Sural 1 x 3 4 2 510 Sach

In the following exercises use summation properties and formulas to rewrite and evaluate the sums 20 8 100 k 5k 1 k 1

culate the probabilities based on the following table Positive Result 2569 478 Virus Present Virus Not Present Total Negative Result 305 3001 Total a What is the probability of having a false positive Convert to percent then round to 2 significant figures DO NOT INPUT THE SYMBOL b What is the probability of having a false negative Convert to percent then round to 2 significant figures DO NOT INPUT THE S SYMBOL c What is the probability of a negative result or a virus not present Convert to percent then round to the one whole number DO NOT INPUT THE SYMBOL

4 What is the total amount owed from a 756 980 loan taken out for 3 years a Simple 2 b Compound

4 1 Suppose you take out a 2500 loan at 7 6 interest for 8 years How much interest will you pay if b The interest was compounded annually a The interest was simple

a s N nova Homework C255 Question 5 of 10 1 point Question Attempt 1 of 1 y 1 2 6 Online Notepad T The graph of f x Oy y 2 3 x 4 6x 7 4 X will behave like which fund 5 S

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Next question For the given simplex tableau a list the basic and the nonbasic variables b fi basic feasible solution determined by setting the nonbasic variables equal to 0 and c decide whether this is a maximum solution a What are the basic variables Use a comma to separate answers as needed X1 X2 X3 S1 S2 Z 0 1 0 3 0 1 0 0 0 3 728 2 4 8 1 0 26 0 16 1 15

10 Given this graph of the function f x Find a f 1 b f 0 c Domain I d f 1

2 Solve for all solutions of x 1 Point tan x 1 Ox k k Z X O x ke Z 4 Ox 0 kez O x 1 2 k k Z x 2 k k Z 37 4

What is the asymptote of the exponential function f x 2 5 4 x 2 y 5 y 4 X 5

SHORT ANSWER Write the word or phrase that best completes each statement or answers the question Solve triangle ABC If necessary round the answer to the nearest tenth 1 A 26 29 m 31 V

1 5 Tranformations 54 543 ys w 237 5 q Complete an equation for the function graphed above Enter abs x for

In some cases it is desirable to decompose a function to write it as a composition of two simpler functions Example 13 Write f x 3 5 22 as the composition of two functions Show solution Solution We are looking for two functions g and h so f x g h z To do this we look for a function inside a function in the formula for f x As one possibility we might notice that 5 is the inside of the square root We could then decompose the function as h x 5 r g x 3 z We can check our answer by recomposing the functions g h z g 5 3 5 2 Note that this is not the only solution to the problem Another non trivial decomposition would be h x and g x 3 5 z Write f x 199 h x 10 Exercise 1 4 8 3 as the composition of two simpler functions f x g h x x 2 20

Consider the function f x 2x 24x 120x 4 10 x 3 This function has an absolute minimum value equal to and an absolute maximum value equal to