Limits & Continuity Questions and Answers

For y f x 3x find f x Ax f x Ax f x Ax f x Ax given x 5 and Ax 4

Graph the function considering the domain critical points symmetry regions where the function is increasing or decreasing inflection points regions where the function is concave upward or concave downward intercepts where possible and asymptotes where applicable f x 3x In x Choose the correct graph below OA B Com C D

Use continuity to evaluate the limit lim sin tan cos 0 0 3 2

2 a Find the value of the limit given in the figure below y 4 e lim x y 2 4 x y xy 4x y 4 x x 1 b 2 c 1 2 d 2 0 L 4x

1 Let f x Inz Determine the following limit algebraically f 27 27p 9p p 27 4pq lim lim P 0 9 40 12 3 p q 12 3 q 16pq

Find the limit lim cos t cos t t In t t 0

5 Solve graphically using the Cartesian Plane f x x 2 x 4 Need to know how this gets solved and then graphed

Question 7 Evaluate the indefinite integral 3dx x In 8x ST

Find the limit lim t 0 6et 6 t 1 t 1 t 2 1 t

Find 0 5T 2 110 2 Round to the nearest hundredth

27 12 There can be asymptotic normality even if there are no moments at a Construct a simple example

The number of hours of daylight on the summer solstice the 172nd day of the year is 15 3 and the number of hours of daylight on the winter solstice the 355th day of the year is 9 1 Complete the function below to model this data Hint The period is 365 days 172 15 3 355 9 1 y cos x O Enter

Use a Table of Integrals to evaluate the integral Jeza sin 6xdx

A mouse wanders through a maze 3 feet east 4 feet north 2 feet east and 5 feet north How far is the mouse from its starting point 12 4 5 x ft 3 Round to the nearest tenth

Use the graph of the function g shown to estimate the function value g 0 2 Find the function value Select the correct choice below and if necessary fill in the answer box to complete your choice OA g 0 2 Type an integer 2 4 2 Ay

Question 19 1 point Litig O 00 8 ol Use the graph of f x to answer the following f 0 f 3 7 f 9 004 O For what value of x does f x 2 5

A ferris wheel is 40 meters in diameter and boarded from a platform that is 4 meters above the ground The six o clock position on the ferris wheel is level with the loading platform The wheel completes 1 full revolution in 4 minutes The function f t Acos B t after the wheel begins to turn How High are you off of the ground after 2 minutes What is the Amplitude A 20 What is the Midline y 24 What is What is the Period 4 That means B T 2 21 D gives your height in meters above the ground t minutes Now please write the function for the height f t 20 cos T 2 meters minutes max min meters meters

Simplify for x in the third quadrant sin cos 1 X Enter the appropriate trig function Enter

Solve the equation and write the solution set x 5 16 0 If there is more than one solution list them in order from smallest to largest Do not enter any spaces in the answer The solution set is

Graph on the number line and express in interval notation The set of all real numbers between 2 and 4 including both 2 and 4 Choose the correct graph below O A O C 6 5 4 3 2 6 5 3 4 1 2 1 0 0 1 2 3 1 20 2 3 4 4 5 5 6 6 OB OD 6 4 5 4 6 5 4 3 3 2 1 0 1 2 1 0 1 Which of the following represents the set of all real numbers between 2 and 4 including both 2 and 4 expressed in interval notation OA 2 4 OD 2 4 OB B 2 4 OC 2 4 2 2

Find f x 1 when f x 4x x 5 To get full points show your work for this problem on your scratch paper

a the limit lim y 4 x y 2 4 x y xy 4x 4x af b Use chain rule to evaluate du if x y x xy y and x u sin v at u 1 v y u cos v 10p 1

vFind the equation of the line in slope intercept form tangent to fat x 1

Match the description with each form of a line Slope Intercept Point Slope Standard Form Horizontal Line A Ax By C 0 B y y m x x C y b D y mx b

nd the equation of the tangent line to the graph of f x at the given point f x 27x 54 at 1 9

se the given graph of the derivative f of a continuous function f over the interval 0 9 to find the following ya X 2 0 X 2 y f x 2 a On what interval s is f increasing Enter your answer using interval notation 4 On what interval s is f decreasing Enter your answer using interval notation X 6 8 x b At what value s of x does f have a local maximum Enter your answers as a comma separated list At what value s of x does f have a local minimum Enter your answers as a comma separated list c On what interval s is f concave upward Enter your answer using interval notation On what interval s is f concave downward Enter your answer using interval notation d What are the x coordinate s of the inflection point s of f Enter your answers as a comma separated list

Divide 4x4 11x 3x 12 x 2x 3 Write your answer in the following form Quotient 4x4 11x 3x 12 x 2x 3 X 0 2 X 2x 3 Remainder x 2x 3 X X

What is f x given the integral function f x cos u du f x sin x o f x cos x sin x O o f x cos x o f x cos x O 34 35 36 37 38

What is the smallest angle of this triangle 12 23 34 Round to the nearest hundredth Ente

Calculate the exact value of sin 67 using the half angle formula Rationalize the denominator if necessary sin 67 5 Enter or Ent

Given the angle 315 a Convert it as radian b Find its coterminal angles in degrees within 720 360 degrees and degrees Enter positive value first and Negative second c Draw the angle below Click the center of the circle then click the correct dot on the circumference 14 14 12 a 0 0 8 06 04 0 2 08 06 04 02 0 2 0 4 0 6 0 8 to 12 0 2 0 4 0 6 0 8 Q 12 14 24

Find the displacement of an object on the interval 2 t 5 if its velocity is shown O 8 5 O 10 O 10 5 O 14 5 KO in an

Problem 1 Use an appropriate test to determine whether the series converges Ek 1 COS 14

If 27 cm of cardboard is available to make a box with an open top and a square base what is the largest possible volume of the box A C 27 10 B cm 2 D cm 24 5 18 5 cm 3 3 cm E None of the above

The function s t 3t 8 gives the distance from a starting point at time t of a particle moving along a line Find the velocity and acceleration functions Then find the velocity and acceleration a t 0 and t 2 Assume that time is measured in seconds and distance is measured in centimeters Velocity will be in centimeters per second cm sec and acceleration in centimeters per second p second cm sec The velocity function is v t Simplify your answer

x y x y TAILS Find the points of inflection of the graph of the function Round your answers to three decimal places If an answer does not exist enter DNE f x 4 sin x sin 2x 0 2 x y 3 4 031 concave downward smallest x value 1 MY NOTES largest x value Describe the concavity Enter your answers using interval notation Round your answers three decimal places If an answer does not exist enter DNE concave upward

2 1 1 1 2 O f 2 3 A x intercept of 2 0 domain of 0 75 Match each transformation of function f with a feature of the transformed function 5 j x h x 3f x 3 vertical asymptote of x 3 X f x 3 g x f x 3 range of 3

The graph of g is given below w Notice this is the graph of the derivative of g not the graph of g itself Answer the following questions about the graph of g based on the graph of g above A State the x values over which g is positive B State the x values over which g is negative C State the x values ver which is zero D State the Y values over which g is undefined

Find all solutions of the equation in the interval 0 2 2x 1 x 2 0 3 2 sin Write your answer in radians in terms of If there is more than one solution separate them with commas x 0

The graph shows the temperature T in degrees Fahrenheit of molten glass t seconds after it is removed from 76 0 1500 a Find lim T 0 3 F What does this limit represent O This is the natural temperature of glass O This is the initial rate at which the glass is cooling O This is the temperature of the kiln This is the temperature of the room b Find lim T 20 F What does this limit represent O This is the natural temperature of glass O This is the initial rate at which the glass is cooling O This is the temperature of the kiln

Find lim h x if it exists X18 f x 4x 4x a b lim h x X48 2x 1 h x c h x lim h x X48 h x lim h x 848 Need Help f x X f x 2 X f x x3

Density Problems Part II Calculate the density of the following different sized blocks Do not forget to include units Recall Volume Length x Width x Height Blocks are drawn to scale Mass 10 g Mass 40 g 1 2 3 Density Density Density Density Mass g Volume cm Object B Mass g Mass 50 g Volume cm Object C Mass g Volume cm Object D Mass g Volume cm Object E Mass g Volume cm Mass 100 g Mass 125 g 3 4 1 1 2 1 Density Density Graphing Part II For each sample use the data below to 1 determine the density 2 determine if the object will sink or float 3 plot the objects mass versus volume on the graph paper Note the graphed line is that object density YOU WILL HAVE 5 DIFFERENT LINES ON THE SAME GRAPH YOU MUST LABEL EACH ONE Object A 4 1 4 8 6 8 Sample Sample Sample Sample 1 2 3 4 2 8 16 32 16 4 2 2 4 2 8 2 12 16 3 3 5 8 4 6 16 4 Density Mass 30 g 7 Density Mass 30 g 8 Density Mass 50 g 24 32 4 4 16 8 32 8 Density g cm 0 5 g cm 3 Sink or float FLOAT Recall the density of water is 1 0 g cm A density greater then 1 0 sinks in water while a density less then 1 0 float

5 Which of the following is the largest domain of the function f x y In 2x y

4 Find inverse of the matrix 133 4 A 1 1 3 3 4

lim x y 1 1 A 1 x xy 2x xy y B 0 c C D 3 E

Solve the following linear programming problem using the Simplex Method If there is no optimal solution explain whether the feasible region is empty or the objective function is unbounded Maximize p 7x 8y subject to x y 3 x y 3 x y 0 Caution Any method other than the Simplex Method will not be graded

Consider the function 1 x 2 x 1 x 3 x 5 defined f over the closed interval 2 4 A Determine all zeroes of f x B Determine all intervals over which f x is decreasing C Consider another function gx ecos x defined over the same closed interval 2 4 Identify all intervals for which g x f x f BIUS X x Q

58 Sketch a graph of a function that satisfies all the conditions that follow lim f x 3 lim f x 1 lim f x 1 lim f x lim f x 1 f 0 0 4 0 8118 110 1 0 1 4

Determine whether f x is continuous at x 3 or not Why or why not cos x 5 f x 2 2 2 3 x 3 3

61 Determine whether f x is continuous at x 2 or not Why or why not x 2x 1 2 5 f x x 2