Limits & Continuity Questions and Answers

tion f x x3 2 is one to one he inverse of f and check the answer he domain and the range of f and f f f and y x on the same coordinate axes x 2 your answer Use integers or fractions for any in the expression domain of f is 0 00 ur answer in interval notation ge of f is ur answer in interval notation

astontbooco 65 What happens to the graph of the function y f x if you make these changes to its equation a replace x with x 8 b replace y with y 2 c replace x with x 4 and y with y 7

Find the domain and range of the function f x 3x 18x 34 O A Domain 00 00 Range 7 00 OB Domain 0 00 Range 0 0 OC Domain 0 00 Range 0 3 O D Domain 00 00 Range 0 00

Graph the piecewise function to find the domain and range Give the answers in interval notation f x 2x 3 if x 0 3x 4 if x 0 Domain inf inf Range help intervals help intervals

3 Answer Write the definitions of sin 0 andcos 0 using the limit definition of derivative de 2512 X126

determine which function matches the graph Which function matches the graph OA f x 2x 2x 2x 2x 2 OB f x 2x 2x 2x 2x 2 OC f x 2x 2x 2x 2 OD f x 2x 2x 2x 2 40

Give the equation of the function whose graph is described The graph of y x is shifted 3 units to the left This graph is then vertically stretched by a factor of 5 and r across the x axis Finally the graph is shifted 7 units downward OA y 5 x 3 7 OB y 5 x 3 7 O c y 5 x 3 7 OD y 5 x 7 3

Scribe Alara shitag Xinyi Boys 7 Short of a formal proof how could you convince someone that lim h 0 8 Apply the limits above to find sin 0 and cos 0 Answer Alara Shitao Baji Scribe Xinyi Aleve Shides Bezi two J sint VI SIVE sin h h O f1 49 1 and lim h 0 hi c nh h Sin la ht h 0 1 h 6 suth k hae ni coshti sah 40

Graph the piecewise function to find its domain and range Your answers must be in interval notation Domain Range help intervals help intervals f x x 1 4 2x if x 3 if x 3

Graph the piecewise function to find its domain and if x 3 if x 4 f x 2x 1 x Domain Range range help intervals help intervals

18 10 140 10 10

Find the domain of f x 8 x 1 Enter the solution in interval notation A help intervals x 1

For f x x 6 and g x x 4 find the following composite functions and state the domain of each a fog b gof c fof d gog a fog x Simplify your answer

Suppose that Find f x f x f x h f x 5hx 7h 6

3 3 points For each function below find the limit as x y approaches 0 0 or show i does not exist a f x y b f x y c f x y Hint try a path of the form y xP for a particular p x6 y x y 2y 9x 9y 2 x 3y sin y y

Graph the function f x 6 sin x y 5 5 0 7 y

Graph the function f x 6 S sin x SH O 7L

Graph the system of linear inequalities Tell whether the graph is bounded or unbounded and label the corner points x 0 y 20 4x y 20 x 4y 20 Use the graphing tool to graph all four inequalities in the system Then use the paint can icon to fill in the region that solves all of the inequalities Click to enlarge graph 22 20 18 16 44 12 10 8 6

Solve the inequality algebraically x 20 x 20 X 0 Move all terms to the left side of the inequality and then define the left side of the inequality as f x Then list the intervals determined by the r undefined Complete the following table Type your answers in interval notation Use ascending order Interval Sign What is the solution 20 Negative 20 0 Positive 0 20 Negative 20 00 Positive Simplify your answer Type your answer in interval notation Use integers or fractions for any numbers in the expression

Graph the function f x X in the standard viewing window of a calculator and trace from left to fight along a representative portion of the interval o 0 Then fill in the blank of the following sentence with either increasing or decreasing Over the interval specified this function is Graph the function Choose the correct graph below A B 10 10 AAALAALLPAALLLLILL 10 10 G 10 10 DINCELALL LLLLLL 10 Q 5 O C 10 10 Q 5 OD 10 10 10

39 42 Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval 39 x x3 0 1 2 40 x 1 x 0 1 41 cos x x 0 1 42 sin x x x 1 2

K Find the equation of the line that passes through the origin perpendicular to 3x y 7 Write the equation in slope intercept form The equation is y Simplify your answer Use integers or fractions for any numbers in the equation Do not factor

For the linear function f x f 4 a f 2 1 Type an integer or a simplified fraction DA 4 a evaluate f 2 and f 4 b find the zero of f and c graph f How can the graph of f be used to determine the zero of f 1 2 Type an integer or a simplified fraction b The zero of f is 2 Type an integer or a simplified fraction c Choose the correct graph below A Q OB A Q Q G How can the graph of f be used to determine the zero of f OA Determine the zero of f by finding the x coordinate of the paint where the line intersects the x axis OB Determine the zero of f by finding the y coordinate of the point where the line intersects the y axis OC Determine the zero of f by finding the x coordinate of the point where the line intersects the y axis O D Determine the zero of f by finding the y coordinate of the point where the line intersesits the x axis DU OD Q

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

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

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

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

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

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