Limits & Continuity Questions and Answers

Answer the following True or False If f x is a differentiable function at x 4 then f x is continuous at x 4 True False

The graph below is the function f x 5 4 3 2 1 21 2 234 2 3 4 5 Determine which one of the following rules for continuity is violated first at x Of a is defined lim f x exists HIG lim f x f a

Evaluate the function at the given values of x Round to 4 decimal places if necessary f x 6 Part 0 4 Part 1 of 4

In a tidal river the time between high tide and low tide is 6 2 hours At high tide the depth of the water is 23 4 feet while at low tide the depth is 11 6 feet Assume the water depth is a trigonometric function of time and assume low tide occurs at noon t 0 A boat requires a depth of 14 feet to sail and is docked at noon If the captain wants to leave the dock as soon as possible how many hours after noon can the boat set sail Give your answer as a decimal rounded to 2 places after the decimal point

The graph of a piecewise defined function is given Write a definition for the function that best describes this graph f x if 4sx 2 2 10 0 2 2

42 Write one pair of polar coordinates that name the point with coordinates E 200 restricting so that 360 8 360 What is one of the answers A 8 160 B 8 110 OC 8 20 8 20 D

49 Change y 4x from rectangular to polar coordinates O A 8 tan 1 B r sin 8 4 C 8 tarr 4 D rcos 8 4

30 Change 3 1 to polar A trigonometric form 2 cos 30 i sin 30 B 2 cos 30 i sin 30 C 2 cos 60 i sin 60 OD 2 cos 60 i sin 60

Question 15 Find domain of the function gi y x 5

What will a 190 000 house cost 9 years from now if the price appreciation for homes over that period averages 7 compounded annually The future cost of the house will be Do not round until the final answer Then round to the nearest cent as needed

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8 of the cylinder x y 25 0 z 3 varies with the distance r from the 2 axis delta 5 r mathrm g mathrm cm 3 Find the mass of the cylinder assuming x y z are in cm mass Include units

If 7000 dollars is invested in a bank account a Find the amount in the bank after 12 years if 17627 19 S Find the amount in the bank after 12 years if 18109 49 Find the amount in the bank after 12 years if 18223 73 S

Noa rides her bicycle for 3 hours and is 32 miles from her house After riding for 9 hours she is 92 miles away What is Noa s average rate during her trip miles per hour

4x x 5 is one to one a Find its inverse and check your answer b Find the domain and the range of f and f The function f x 5x 4 x b Find the domain of f Select the correct choice below and if necessary fill in the answer box to complete a f x Simplify your answer A The domain is xx 5 Type integers or fractions Use a comma to separate answers as needed B The domain is xx Type integers or fractions Use a comma to separate answers as needed C The domain is xjxs Type integers or fractions Use a comma to separate answers as needed D The domain is the set of all real numbers Find the range of f Select the correct choice below and if necessary fill in the answer box to complete you OA The range is yly z Type integers or fractions Use a comma to separate answers as needed OB The range is yly Type integers or fractions Use a comma to separate answers as needed C The range is ylys Type integers or fractions Use a comma to separate answers as needed

4x x 5 a Find its inverse and check your answer b Find the domain an The function f x is one to one 5x a f x Simplify your answer 4 X b Find the domain of f Select the correct choice below and if ne A The domain is xx Tune intonom r fractions Use a comma to separate ansv

Data was collected for 40 randomly selected trees growing in the median strip of a busy freeway The height of each tree in inches was recorded Data was recorded to the nearest whole inch The data is summarized in the histogram below Frequency 225642 12 10 8 10 15 20 25 30 35 40 45 50 height in a Based on this histogram determing the minimum and maximum possible tree heights for the recorded data Note Each class contains its lower class boundary but not its upper class boundary Minimum tree height inches

The graph of y f x solid and its tangent line at a 0 dashed is shown a second point on the tangent line is also marked Assume f x is continuous 10 2 a NUM 15 EU 10 9 5 f x z 0 5e 5 37 10 N 7 5 9 10 lim 20 2

Write an equation for the function grap 7 6 5 4 3 2 4 Sn ha My CN CY

Use the Comparison T CO 2 n 1 7 n 9 O diverges O converges

Let f x 4x 7x 3 According to the definition f x lim h 0 Enter the limit in reduced form

Let f x 2x 4x 3 Expand and simplify f x h f x h

Let f x According to the definition 3x 2 f x lim h 0 f x enter the limit in reduced form enter the limit

5 4 3 2 1 4 3 f 2 2 7 2 3 4 1 2 3 4 Use the graph of f x above to estimate the value of f 2 to one decimal place

In a tennis club there are 5 boys and 3 girls in a training squad Two are chosen at random to represent the club Determine the probability that they are 3 marks Both boys Both girls c A boy and a girl a b

2 marks a a Monday What is the theoretical probability as a fraction that the next person you meet was born on b a weekday

8 Recall we say that function g dominates function f provided that f x lim f x o lim g x o and lim 0 g z 2 0 00 80 a Which function dominates the other In x or T b Which function dominates the other In x or n can be any positive integer c Explain why e will dominate any polynomial function d Explain why will dominate In x for any positive integer n e Give any example of two nonlinear functions such that neither dominates the other

5 Let f and g be differentiable functions about which the following information is known f 3 g 3 0 f 3 g 3 0 f 3 2 and g 3 1 Let a new function h be given by the rule h x f On the same set of axes sketch possible graphs of f and g near x 3 and use the provided information to determine the value of lim h x I 3 Provide explanation to support your conclusion

3 Determining if L H pital s Rule applies Compute the following limits using l Hopital s rule if appropriate Use INF to denote and MINF to denote 1 cos 7x cos 3x lim 201 lim I 1 4 3 1 x2 1 H

4 Using L H pital s Rule multiple times Evaluate the limit using L Hopital s rule 3 15x lim I 0 e21

2 L H pital s Rule to evaluate a limit In x 4 Find the limit lim 14x 16 Sommer Enter undefined if the limit does not exist Activate

Question 9 o 2 A one to one funct t x x 3 x 8

A monopolist faces the following demand curve Quantity 3 4 5 O 1 11 9 Price 7 15 12 Suppose the monopolist has total fixed costs equal to 5 and a variable cost equal to 4 per unit for all units pro What is the total profit if she operates at her profit maximizing price 9 6 3

MTH 161 Quiz C2S5 Question 1 Let f x 2x 3 x 1 as x infinity f x

Select all values of a where the function defined by the following graph is not differentiable sices select all that apply x 5 x 4 10 x 3 5 2 2 y 104 5 To 0 5 10 2 1 5 x 0 10 X 2 1 x 3 x 4

11 10 9 8 7 6 5 4 3 2 T f x 2 1 2 3 12 10 7 8 a The curve above is the graph of a sinusoidal function It goes through the points 9 2 and 1 2 Find a sinusoidal function that matches the given graph If needed you can enter 3 1416 as pi in your answer otherwise use at least 3 decimal digits com

7 DO 4 of these 5 Show alged f x 5ex 2 find f x Show algebra steps f x 3x 6 and g x x 1 Find g f x Simplify g x 4x 2 6 How does this graph compare to f x 4 List what the transformations are compared to the parent function Simplify condense 2logx 5logy logw Expand In 2 y 03 1 2

2 What is the Domain and Range of g x log3x Domain Range BBWANA

If f x StartRoot x minus 3 EndRoot which inequality can be used to find the domain of f x

Evaluate the indicated limit by evaluating the function for values shown in the table and observing the values that are obtained Do not change the form of the function 6x 4 10x 2 lim X18 10 100 8 Type integers or decimals rounded to four decimal places as needed f x 1000

For the function shown in the graph to the right find the following a f 2 b lim f x x 2 a Select the correct choice below and if necessary fill in the answer box to complete your choice OA 2 OB 2 does not exist GITS

Name Go down deep enough into anything and you will find mathematics Dean Schlicter SHOW WORK ON EACH PROBLEM FOR ANY PARTIAL CREDIT 10 pts each 5 Determine the limits using techniques from class not a calculator 1 cos x a lim x 0 x2 Optimization Newton s Method l Hospit b lim x ex x118 c lim cos x x r 0

4 Before rounding c a 20 82 b 13 12 c 23 12 d 15 28

3 b a b C d 32 sin 77 sin 32 32 sin 71 sin 77 32 sin 77 sin 71 32 sin 32 sin 77

There exists three different values of n for which the expansion of x n 3x where n EN has an x term Which value of n gives a negative coefficient for the x term

Determine whether the series is convergent or divergent If it is convergent find its sum If the quantity diverges enter DIVER n 1 6 n n 1

Determine whether the series is convergent or divergent by expressing s as a telescoping sum If it is convergent find its sum If the quantity diverges enter DIVERGES n 1 18 n n 3

4 We solve t Y 0 0 dy t y numerically Fill in the blanks below dt 1 2 3 4

17 Evaluate log 18 What is the answer a 1 8343 b 3460 c 2 8904 d 5452

20 Write log b 3 4 2 5 7 35 a 2 log 3 4 log 5 7 log 35 log2 3 4 log 5 7 log 35 c log 2 3 4 log 5 7 log 35 2 log 3 4 log 5 7 log 35 d as an algebraic sum of logarithms