Limits & Continuity Questions and Answers

K Find the absolute extrema if they exist as well as all values of x where they occur for the function f x x 12x 36x 1 on the domain 9 0 www Find the derivative of f x x 12x 36x 1 f x Identify the absolute maximum if it exists as well as all values of x where it occurs Select the correct choice below and if necessary fill in the answer boxes to complete OA The absolute maximum is which occurs at x Round the absolute maximum to two decimal places as needed Type an exact answer for the value of x where the maximum occurs Use a comma to separate ar as needed OB There is no absolute maximum Identify the absolute minimum if it exists as well as all values of x where it occurs Select the correct choice below and if necessary fill in the answer boxes to complete y OA The absolute minimum is which occurs at x Round the absolute minimum to two decimal places as needed Type an exact answer for the value of x where the minimum occurs Use a comma to separate ansv as needed B There is no absolute minimum

QUESTION 8 Solve ABC subject to the given conditions Round the lengths of sides and measures of the angles to 1 decimal place if necessary A 10 6 C 98 8 b 53 OB 70 6 a 10 3 c 54 4 OB 70 6 a 10 7 c 55 3 B 70 6 a 10 7 c 54 4 OB 70 6 a 11 5 c 61 8

Find the exact value 1 sin 1 sin 1 2 O NY WN WH 2 2 3

Graph of a function y f x is shown What is lim f x x 1 Enter a number only 5 4 3 y f x 1 3 1 2 1 1 2 3

Estimate the slope of the tangent line to the curve at the given point x y The slope of the tangent line is approximately Type an integer or a fraction COOR

Use the properties of limits to help decide whether the limit exists If the limit exists find its value lim X 6 2 x 36 X 6 O A Select the correct choice below and if necessary fill in the answer box within your choice x 36 X 6 lim X 6 Simplify your answer OB The limit does not exist and is neither o nor o www

Use the graph to find the following limits a lim f x b lim f x 71 x 1 a Find lim f x or state that it doesn t exist Select the correct choice below and fill in 2 1 any answer boxes in your choice OA lim f x x111 Type an integer OB The limit does not exist b Find lim f x or state that it doesn t exist Select the correct choice below and fill in any x 1 answer boxes in your choice OA lim f x x 1 Type an integer Ay 2

Which statement correctly describes the function f x 6x 5 if x 0 2x 10 if x 0

Let C x be the cost to produce x batches of widgets and let R x be the revenue in thousands of dollars Complete parts a through d below 2 3 273 2 R x 5 x 10x C x 5x 5 a Graph both functions Identify the vertex of R x The vertex of R x is Type an ordered pair Simplify your answer Type integers or decimals

answer 10 5 5 10 5 Selecting an option will display any text boxes needed to complete your answer Increasing on one interval Decreasing on one interval Constant on one interval Increasing on one interval and Decreasing on another Increasing on two intervals Decreasing on two intervals 10

34 O Hide Transcribed Text 4 Calculate the limit operatorname Lim rightarrow infty left sqrt frac 2 n

Suppose the functions f x and g x are both twice differentiable Suppose also it is known that f 3 2 f 3 1 g 3 1 and g 3 4 xf x 6 g x 1 Then lim x3

Calculate each of the following limits 9 9 lim 4x 8x 3 85 Using synthetic division select the 680x y 10 8 5 lim x 7x 10 x 2 56x y 11 1

Calculate each of the following limits 9 9 85 lim 4x 8x 3 10 8 5 lim x 7x 10 x x 2

x y 90 A r 6x y 15x y 20x y 15x y 6xy y B 15x y 15x y y C D E x 2x y 6 x 6x y 30x 60x y 30xy 6xy yo x 6xy 15x y 20x y 15x y 6x y yo

A woman deposits 15 000 at the end of each year for 14 years in an investment account with a guaranteed interest rate of 4 compounded annually a Find the value in the account at the end of the 14 years b Her sister works for an investment firm that pays 3 compounded annually If the woman deposits money with this firm instead of the one in part a how much will she have in her account at the end of 14 years c How much would she lose or gain over 14 years by investing in her sister s firm a The woman s deposits form an should be used because the deposits are made at the of each period Therefore the formula

MY NOTES ASK YO Find the interest rate needed for an investment of 4 000 to grow to an amount of 4 400 in 5 months if Interest is compounded monthly Round your answer to the nearest hundredth of a point

Solve the following system of equations Enter your answer as an ordered pair 3x y 16 2x y 9 Question Help Video 1 Video 2 Video 3 Message instructor

Solve this system of equations 8 2x 2y x 2y 9 O No solution O Infinite number of solutions

dentify the mi n m ax value of each Then sk f x x 2 8x 20

Question 11 2 points What is the range of the following function g x log x 3 All Real Numbers R A 54 3 2 1 to S 3 21 4 2 12 3 N

Use the method of undetermined coefficients to solve the given system x x 3 t 4 X t

problem you will calculate the area between f x 3x and the x axis over the interval 0 2 using a limit of right endpoint Riemann sums Ax Express the following quantities in terms of n the number of rectangles in the Riemann sum and k the index for the rectangles in the Riemann sum a We start by subdividing 0 2 into n equal width subintervals ro 1 1 2 Tn 1 En each of width Az Express the width of each subinterval Az in terms of the number of subintervals n 1 2 3 x b Find the right endpoints 21 22 23 of the first second and third subintervals 0 1 2 2 3 and express your answers in terms of n 9 Enter a comma separated list c Find a general expression for the right endpoint x of the kth subinterval 1 where 1 k n Express your answer in terms of k and n 94 d Find f x in terms of k and n f x e Find f x Ax in terms of k and n f xk Ax f Find the value of the right endpoint Riemann sum in terms of n f x Ax k 1 limf z Ar k 1 g Find the limit of the right endpoint Riemann sum 121400 4 Area lim 11 900 1 zn Az 8

Express 341000 m in scientific notation preserving the correct number of significant figures 3 4x105 m 3 41x105 m 3 41000x105 m 34 1x104 m

Points 0 of 1 Which loan best meets the person s needs Round to the nearest cent as needed Someone needs to borrow 13 000 to buy a car and the person has determined that monthly payments of 250 are affordable The bank offers a 4 year loan at 8 APR a 5 year loan at 8 5 or a 6 year loan at 9 APR Which loan best meets the person s needs Explain OA The second loan best meets the person s needs because the monthly payment of maximum budgeted amount of 250 per month OB The third loan best meets the person s needs because the monthly payment of budgeted amount of 250 per month Save OC The first loan best meets the person s needs because the monthly payment of budgeted amount of 250 per month OD None of the loans meet the person s needs is less than the is less than the maximum is less than the maximum

Suppose 6x ax 4x f x x ax 12 for x 2 and near 2 Determine a in such a way that the Squeeze Theorem can be used to find lim f x The x 2 provided For the toolbar press ALT F10 PC or ALT FN F10 Mac BIUS

10 5 pts Use the Squeeze Theorem to show lim x 3 sin x 0 logical steps 0 Make sure your work has specific

2 5 pts Suppose y f x is defined for all x Sketch a graph of y f x with the properties it is discontinuous at x 2 but continuous elsewhere with lim f x 1 x 0 y 9 8 7 6 5 4 3 9 8 7 6 5 4 3 2 1 2 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 678 9

5 Use the delta epsilon definition of a limit to prove lim 24 X 2

3x 4 x 2x 5 12 Prove using the mathematical definition lim MIN 3 Hint definition involves x N

8 Determine the indicated limits Indicate o or oo for those limits that do not exist Describe division by small positive or negative numbers in your work 2x 7x 4 a lim x 2 x x 2 2x 7x 4 x 1 x x 2 b lim 2x 7x 4 x 1 x x 2 c lim

7 consider the function f x x sinx x tan x if x if x T Determine if f x is continuous at x Provide mathematical proof the three criteria for your answer If it is discontinuous what type of discontinuity is it nge

Chapter 2 RE Test Limits Name SHOW WORK FOR ANY CREDIT 10 points per problem 2 10 5 pts 1 Algebraically determine the indicated limits Indicate too or oo for those limits that do not exist 1 1 atha a 0 c lim a lim b lim h 0 x 4 1 x 3

Simplify to a single trig function with no denominator sec 0 csc 0

Convert as indicated 99 kg to lb Click the icon to view the equivalent measures table 99 kg lb Round to the nearest hundredth

Find the midpoint of the line segment joining the points R 2 3 The midpoint is Type an ordered pair Use integers or simplified fractions for any numbers in the expression

10 C 6 4 2 1 9 4 8 7 1 1 1 1 I 1 1 1 1 1 1 1 6 1 1 V 5 4 3 104 9 8 7 6 5 32 1 OF 2 3 2 10 2 1 2 3 Use the graph to fill in the table below If a y value is undefined or a limit does not exist type DNE lim f x lim lim f x f c Is f x continuous at x c I C I C 4 5 6 7 8 9 10 Select an answer Select an answer No the limit does not exist No the y value does not equal the limit No the y value is not defined Yes Select an answer Select an answer Select an answer

8x 5 if x 6 4x b if x 6 Find the value of b so that f x is continuous Suppose f x b

Let f x lim f x I 1 10 I 1 f 1 35 I 8 Compute the quantities below Write DNE if the limit does not exist or the value is undefined lim f x I 1 if a 1 if x 1 Since the above three quantities are not all defined discontinuous at x 1 we know that f is List ALL numbers at which f is discontinuous Be sure to check the functions defined to the left and right of 1 for discontinuities

www Give the intervals where the function is continuous Answers are expected in interval notation e g 00 2 U 2 00 Use U between intervals and use and oo for negative and positive infinity and are differentiated Unless explicitly shown in the ima assume the left and right endpoints are finite and closed

1 point If f x e x g x 8x 8 and h x x Find a simplified formula for the function below f g x h x

Perform the conversion Round to the nearest hundredth when necessary 16 3 cm to in Click the icon to view the equivalent measures table 16 3 cm in

Convert as indicated 69 mi to km Click the icon to view the equivalent measures table 69 mi km Round to the nearest hundredth

If 41 and 42 are complementary angles and if the measure of 41 is 86 more than the measure of 42 determine the measures of 41 and What are the measures of the angles 41 has measure 42 has measure O O

Given that ZEFG is a right angle find x Type an integer or a decimal X x 17 FO X

A window in Lynn and Jorge s house needs to be replaced The rectangular window is 3 feet 7 inches tall and 2 feet 6 inches wide Change each of the measurements to inches a Find the perimeter of the window in inches b If the perimeter needs to be sealed with insulation that is 0 70 per inch how much will it cost to insulate the perimeter a The perimeter of the window is inches Simplify your answer

Divide in the indicated base 24 2004 The quotient is R4

1 Given f x x 3 answer will be an expression involving h 1 find the average rate of change of f x on the interval 5 5 h Your Question Help Video Message instructor

Find the average rate of change of g x 6x4 9 x3 on the interval 3 4

Add in the indicated base 4537 6467 4537