Limits & Continuity Questions and Answers

A particular country s exports of goods are increasing exponentially The value of the exports t years after 2007 can be approximated by V t 1 4e0 043t where t 0 corresponds to 2007 and V billions of dollars a Estimate the value of the country s exports in 2007 and 2020 b What is the doubling time for the value of the country s exports a The value of the country s exports in 2007 is billion Simplify your answer Round to the nearest tenth as needed Do not include the symbol in your answer The value of the country s exports in 2020 is billion Simplify your answer Round to the nearest tenth as needed Do not include the symbol in your answer b The doubling time is approximately years Simplify your answer Round to the nearest integer as needed

f x 7 what is the domain of f x 6 Given f x 2x 3 x 3 2 x 3 x 3 If f x x and gax x 2 5 what x y point did 0 0 become on If fl2 3 then f 3 f x x 7 find f x Graph f x 3 x 1 x X 2 X 3 10 Find fl4 Find f 3 The graph of g x contams the point a what is the corresponding point b what is the corresponding point 2 1 x 3 on glx glx If 31 is on the graph of f x what g x if glx f x 4 quint

Decide whether the relation to the right defines a function and give the domain and range Does the relation define a function Yos No What is the domain 2 3 4 Use a comma to separate answers as needed What is the range Use a comma to separate answers as needed CD

Exploration 2 1 13 Use Taylor s Theorem to derive the error terms the formulas and xo f xo f xo h f xo h 9 f xo h f xo h 2h error error h 2 f E h

Determine the domain and range of the relation and tell whether the CIP Using interval notation the domain is Simplify your answers Type your answer in interval notation

Use the graph of y f x to find each function value a f 3 c 1 5 a 1 3 b f 0 d f 6 Ay

K Evaluate the function f x 2x 8 at the indicated values a Find 1 3 b Find f 7 a f 3 Simplify your answer

Does the relation define a function Yes No What is the domain Use a comma to separate answers as needed

The accompanying data represent the percentages of teenagers in various countries that have used certain drugs Complete parts a through c below J K D B C D 40 5 20 1 E F G 37 19 22 16 H 1 5 6 4 52 35 3 17 7 14 4 31 23 Country Drug 1 x Other Drugs y A 22 5 a Determine the correlation coefficient between the percentage of teenagers who have used Drug 1 and the percentage who have used other drugs r Round to two decimal places as needed b Find the equation of the regression line for the percentage of teenagers who have used Drug 1 and the percentage who have used other drugs y x Round to two decimal places as needed c What percentage of teenagers can be anticipated to use drugs other than Drug 1 in a country where 10 of teenagers have used Drug 1 Round to the nearest whole number as needed

fis the function defined by f x A B C lim x 2 x 2 z 2 lim x 2 lim 2 x 2 2 lim z 2 2 4 24 2 2 then limf x is equivalent to which of the following z 2

Use the graph of the function to answer parts a through f P x x 5 1 IN The coordinates of the vertex are 5 1 Type an ordered pair b Give the domain and range What is the domain of P x 0 00 Type your answer in interval notation What is the range of P x 1 00 Type your answer in interval notation c Give the equation of the axis of symmetry x 5 Type an equation d Give the largest open interval over which the function is increasing Type your answer in interval notation 10 B

Consider the function f x x 4x 6 on the interval 0 4 Verify that this function satisfies the three hypotheses of Rolle s Theorem on the inverval f x is on 0 4 f x is on 0 4 and f 0 f 4 Then by Rolle s theorem there exists at least one value e such that f c 0 Find all such values c and enter them as a comma separated list Values of c

Express the function G in the form f g There is more than one correct answer The function is of the form y f g x Use non identity functions for f x and g x G x f x g x 1 x 1

Find f g h 0 G f x x 2 g x x h x x 2

Use f x 3x 5 and g x 2x to evaluate the expressio a f f x b g g x

From the graphs of f and g in the figure we find the following Assume that each point lies on the gridlines f g 8 f g 8 fg 8 8 2 1

stone is dropped in a lake creating a circular ripple that travels outward at a speed of 65 cm s a Find a function g that models the radius as a function of time t in seconds g t b Find a function f that models the area of the circle as a function of the radius r in cm f r c Find f g f g What does this function represent This function represents the area of the ripple as a function of the radius This function represents time as a function of the area of the ripple This function represents the radius of the ripple as a function of time This function represents the area of the ripple as a function of time This function represents the radius of the ripple as a function of area

Consider a triangle 4 BC like the one below Suppose that 4 34 b 67 and c 3 The figure is not drawn to scale Solve the triangle Carry your intermediate computations to at least four decimal places and round your answers to the nearest tenth If there is more than one solution use the button labeled or 0 0 a B or X No solution S

Find the area of the triangle below Carry your intermediate computations to at least four decimal places Round your answer to the nearest hundredth 6 in 4 in 36 A in X

4 Plot the following functions a f x x 2x x 1 b g x x sin x 2 tanx 0 x 4 5 x 5

Evaluate the indicated limits 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 a Find lim x b f x I 1 x Find lim I 0 X ex X 1 1 0 900 0 1 0 990 0 01 0 999

Show all your work and clearly box the final an 1 Use the following graph of f x to find each limit 8 pts 5 Find lim f x x 2 c Find lim f x x 2 2 7 d Find f 2 1 2 3 4 a b C 2 Graph the following function to find the indicated limits and function values 13 pts a Find lim f x x 2 d lim f x lim f x S lim f x lim f x X 4 3 for x 2 x 1 for x 2 f x 2x 3 N 65432 1 7 6 5 4 3 2 1 1 123456 2 3 4 5 6 1234567 e Is the function continuous or discontinuous at x 2 Explain your answer using the definition of a continuous function Hint Verify the three conditions we discuss in class 4 pts

Determine the values of x for which the function as represented by the graphs is continuous If the function is not continuous determine the reason a b 2 1 20 1 2

Consider the following function v x Answer if x 2 6 4 x if x 2 Step 1 of 3 Identify the general shape and direction of the graph of this function on the interval o 2 776 X x

Answers parts a through e a Express the volume V of the box as a function of the length x of the side of the square cut from each corner V 225x 60x 4x Simplify your answer b What is the volume if a 2 inch square is cut out The volume is 242 cubic inches Simplify your answer c What is the volume if a 6 inch square is cut out The volume is 54 cubic inches Simplify your answer d Graph V V x For what value of x is V largest Choose the correct graph below an example Q Search K An open box with a square base is to be made from a square piece of cardboard 15 inches on a side by cutting out a square from each comer and turning up the sides See the figure d Graph V V x For what value of x is V largest Choose the correct graph below OA Get more help Xmin 0 Xmax 7 5 Xscl 0 5 Ymin 0 Ymax 275 Yscl 25 For what value of x is V largest X EIN 5 inches B Xmin 0 Xmax 7 5 Xscl 0 5 Ymin 0 Ymax 275 Yscl 25 come OC Q Xmin 0 Xmax 7 5 Xscl 0 5 Ymin 0 Ymax 130 Yscl 10 OD Clear all Q Q 15 m XminEO Xmar 15 Od 0 Ymin 0 Ymax 275 Yad 25

Identify the even vertices and identify the odd vertices LIF Use a comma to separate answers as needed OA The only odd vertex is OB The odd vertices are G Use a comma to separate answers as needed B D OC There are no even vertices Identify the odd vertices Select the correct choice below and if necessary fill in the answer box to complete your choice Ea G

If 9000 is invested for t years at 8 interest compounded continuously the future value is given by S 900000 dollars Complete parts a through e a Graph this function for Osts 15 Choose the correct graph below OB AS 34 000 c Complete the following table t Year S 8 17 068 33 Round to the nearest whole number as needed AS Complete the following table t Year S 5 5 34 000 Q b Use the graph to estimate when the future value will be 20 000 Choose the correct answer below A 19 98 years B te 19 96 years VC 20 Type an integer or decimal rounded to two decimal places as needed AS 34 000 OC t 8 98 years o 5 OD 34 000 00 1

Let f x 3 x If g x is the graph of f x shifted up 4 units and right 4 units write a formula for g x g x Enter as sqrt x

An open box with a square base is to be made from a square piece of cardboard 9 inches on a side by cutting out a square from each corner and turning up the sides See the figure Answers parts a through e d Graph V V x For what value of x is V largest Choose the correct graph below OA Q 5 Xmin 0 Xmax 4 5 Xscl 0 5 Ymin 0 Ymax 60 Yscl 10 For what value of x is V largest x 01 X OB G Xmin 0 Xmax 4 5 Xscl 0 5 Ymin 0 Ymax 60 Yscl 10 O C Q G Xmin 0 Xmax 4 5 Xscl 0 5 Ymin 0 Ymax 30 Yscl 5 X 19 in Xmin 0 Xmax 45 X5 Ymin 0 Ymax s6 ente sa Clear all

Given f x 3x 4 and g x 2x 5 find f g x O A x 1 O B 5x 1 C 5x 1 O D 5x 9

7 B 74 3 a 859 m b 783 m

For each of those 4 values below answer the following questions a Does the limit exist if yes the value b What type of discontinuity is it 11 Examine the graph below and answer the questions that follow assume a scale of 1 paying attention to the values where a discontinuity occurs C4 x 8 a b x 1 2 a b 1x 2 a b x 2

10 The function f x is known to be continuous determine the value of a and the value of b required to make this statement true T4 6 x x 1 f x ax b 1 x 2 5x 3x 2

9 Determine the limit of lim x 8 2 X x 8 T4

5 Determine if the limit exists at x 1 and if so determine the value 2x 3 x 1 f x 6 x 1 3x 2 x 1 K2

4 What is limx 4 x 2x 3 a 3 b 2 c 3 d 21

As x f x As x 5 f x As x 5 from the right f x lim f x x 3 lim f x X 6 graph of f x below 6 What is the equation of the horizontal asymptote What is the equation of the vertical asymptote Classify the discontinuity at x 3 Classify the discontinuity at x 6

f x I 2 kx x 1 5 3x x 1 207 If lim f x 3 and lim g x 7 find lim x 2 x 2 x2 f x 208 Find a value for k so that f x is continuous if g x

206 Find lim 72 3

101 Solve the inequality 1 1 4 x 2 0

109 Find the average rate of change between x 2 and x 5 based on the graph below 11 10 9 6 54321 Ay y f x I 4 5 6 7 8 9 10

Apply L H pital s Rule to evaluate the following limit It may be necessary to apply it more than once 5x 5 x 7 2 lim

4 No calc The x coordinate of the point where the line tangent to the parabola y 8x at 2 4 intersects the axis of symmetry of the parabola is x A 1 B 1 C 2 D 0 5 No calc If y 3sin x 4 cos x then y y A 6sin x 8 cos x B 6sin x 8 cos x C 6sinx 8cos x D 6sin x 8 cos x E 0 6 Calc Let f be the function given by f x tan x and let g be the function given by g x x At what value of x in the interval 0 x do the graphs off and g have parallel tangent lines A 0 B 0 660 7 No calc The lim A40 A 1 C 2 083 h 8 No cale lim A 4 D 2 194 at x 3 is B 0 1 D 3 lim 1370 13 0 h 70 x 16x 20x 4x B 2 I m 0 0 0 0 E 2 5 D 2 E 2 207 E DNE E 4

Graph all vertical and horizontal asymptotes of the rational function 10 x 4 1 X 5

Graph all vertical and horizontal asymptotes of the rational function 9x 4 3x 6

3 Use the graphs of f and g to find the domain of the function indicated a f g b f g c fog 2 f x 4 Figure 1 Graph of f x and g x g x

1 Calculate each of the following and simplify your answer f g fg fog go f a f x x and g x 1 b f x x and g x x 1 X x x 1

104 If lim f x 4 and lim g x 5 find lim 2f x g x

figure shows the graph of a quadratic function y f x AY 20 y f x 16 12 8 16 4 12 4 8 12 16 Q How many real solutions are there f x 1 Choose the correct answer below OA One OB Two OC None

y x 1 1 The whether its graph opens upward or downw Select the correct choice below and fill in the answer box to complete your choice Type an ordered pair OA The vertex of the parabola is OB The vertex of the parabola is and it opens downward and it opens upward