# Limits & Continuity Questions and Answers Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
Does the relation define a function Yes No What is the domain Use a comma to separate answers as needed Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
Find f g h 0 G f x x 2 g x x h x x 2 Calculus
Limits & Continuity
Use f x 3x 5 and g x 2x to evaluate the expressio a f f x b g g x Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
7 B 74 3 a 859 m b 783 m Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
9 Determine the limit of lim x 8 2 X x 8 T4 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
4 What is limx 4 x 2x 3 a 3 b 2 c 3 d 21 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
206 Find lim 72 3 Calculus
Limits & Continuity
101 Solve the inequality 1 1 4 x 2 0 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
Graph all vertical and horizontal asymptotes of the rational function 10 x 4 1 X 5 Calculus
Limits & Continuity
Graph all vertical and horizontal asymptotes of the rational function 9x 4 3x 6 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
104 If lim f x 4 and lim g x 5 find lim 2f x g x Calculus
Limits & Continuity
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 Calculus
Limits & Continuity
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