Limits & Continuity Questions and Answers

Given that g t is continuous find the value of c if g t ct 2 cos if t 2 if t 2 32 A c 1 2 O B c 1 2 O C c 0 D c 2 E None of the above

10 8 6 Choose the correct statement lim f x 18 X 1 2 lim f x x 1 8 6 4 2 0 2 4 6 8 1 2 4 CO 6 8 I 1 19 10

10 8 6 lim f x x 2 4 Choose the correct statement x x 0 2 104 8 9 6 4 2 02 6 8 10 fx 2 4 6 8 10

P7 10 8 6 lim f x 00 X 4 4 Choose the correct statement lim f x 2 8 4 6 8 10 2 4 6 8 10 X

Let F x 2 tan x 3 where x 1 1 2 Find the inverse function F 1 x show work of f and the Domain and Find the range range of

5 4 155 Find the following limits a lim f x b lim f x H112 112 7 Y 3 2 1 0 b 1 2 3 4 c lim f x x112 1 d lim f x 1 3 4 e lim f x H15

Find limx 3 5 2 Remember that x is the greatest integer function x O A 3 5 1 OB 3 5 OC 0 OD 2 3 5 O E Does not exist

Find lima 3 5 2 Remember that x is the greatest integer function x OA 1 OB 3 5 O C 0 O D 1 3 5 2 3 5 OE Does not exist

Does the graph represent a function that has an inverse function Q 10 10

2 A convex lens with focal length fcentimeters will project the image of an object on a point behind the lens If an object is placed a distance of p centimeters from the lens then the distance q centimeters of the image from the lens is related to p and f by the lens equation 14 1 pq f object K p image A If the focal length of the convex lens is supposed to be 5 cm and if the image is formed 7 cm from the lens find the distance from the lens to the object p It s not necessary to simplify your answer B Find an expression that gives q as a function of p assuming that the focal length is a constant of 5 centimeters C Sketch a graph of q as a function of p i e q p assuming that the focal length is a constant of 5 centimeters Show any important features of the graph

Answer the following questions about the sphere whose equation is given by x y 2 8x 4y 80 1 Find the radius of the sphere Radius r O 2 Find the center of the sphere Write the center as a point a b c where a b and care numbers

What is the terminal point of the vector a Answer 7 A a 1 4 based at P 2 1

Find the components and length of the following vectors i 5j Components Length i 5j Components Length 4i 5j Components Length 4i 2j Components Length A B 1 0 2 10 4 P

of the definite integral we have 6k 6km From the definition limn 0 1 sin k 1 n 6 A fo 2x sin 2 x dx 6 O B f x sin x dx 3 O c f x sin 2x dx O D Jo 2x sin 2x dx 6 O E f x sin 2 x dx

Here is a graph of the function g 54 3 2 m s 40 W ea 1 H 2 3 4 2 Use the graph to find the following If there is more than one answer separate them with commas a All local minimum values of g b All values at which g has a local minimum

Graph the solution to the inequality on the number line w 4 7

Factor the following expression 2 3 28475 16xx

The graph of a function g is given a State the values of g 2 g 0 g 2 and g 3 b For what value s of x is g x 3 c For what value s of x is g x 3 d State the domain and range of g e On what interval s is g increasing YA 3 3 0 3 9 x

7 Suppose you know that f x is an odd function on the domain of all real numbers and that th function is concave up on the intervals 0 x 3 and 5 x and concave down on the interval 3 x 5 List ALL intervals on which the function fx is concave up and ALL intervals on which the function fx is concave down explain how you know these to be correct

2 Consider the function f x x x Use graphical methods to discuss the differentiability of the function In particular find all values of x for which the function appears non differentiable and explain why Include at least one graph illustrating each point of non differentiability

1 Below is a graph of the function y f x 5 From your observations of this graph state with reasons the values of x at which fis not differentiable 2 Consider the function f x x x Use graphical methods to discuss the differentiability of the function In particular find all values of x for which the function appears non difforontin

6 Below is the graph of y g x 17 0 4 I Note The open circles at x 2 are to imply that the function g is not defined there Answer the following questions about the function g x A Is is possible impossible or certain that g is continuous at x 2 Explain B Is it possible impossible or certain that g 2 0 Explain C Is it possible impossible or certain that g has a vertical asymptote at x 2 Explain D Is it possible impossible or certain that g has a jump discontinuity at x 2 E Is it possible impossible or certain that g has a removable discontinuity at x 2 F Make a sketch of what the graph of y g x could look like

13 Find the limit of the sequence if it converges otherwise indicate divergence 2 2 n an 14 Find the limit of the sequence if it converges otherwise indicate divergence an n n 20n

Now take a look at the following three functions f x x 2x 1 500 g x x 0 X k x x 21 For each function do the following 1 Compute the derivative that is the derivative function by evaluating the limit for f x h f x example limo h Show your work Notice that for the function k x you ll have to compute the limit differently depending on whether x 2 or x 2 and you ll end up with a piecewise function for the derivative Since the graph of k x is made of straight lines you can also find its derivative by looking at the actual slopes 2 Graph the derivative f x on your calculator by assigning it to the function Y Draw the graphs that you create on your calculator Make sure to label and submit them alon with the rest of the assignment 3 Graph the function Y nDeriv original function X X By doing this you re having the calculator evaluate the derivative numerically at each point and graph the results 4 Discuss how the two graphs are the same or different Zoom in and out to find places where the graphs differ A good idea is to set Y3 Y2 Y deselect but don t erase the functions Y and Y2 and look at the graph of Y3 Discuss what this illustrates and see if it helps you in your comparison Discuss the things you did in order to compare the two graphs After you ve done these things for each of the three functions make a final comment about when the graph made by nDeriv was most or least accurate Try to come up with a general comment such as Dori

Earth s distance from the center of the sun as a function of time 1 Point

4 Suppose f x if x 2 x 3 if 2 x Identify any points of discontinuity and determine giving reasons if they are removab infinite essential or jump discontinuities 6 Points

A First class postage as a function of weight 1 Point

B Atmospheric pressure as a function of altitude 1 Point

1 Given f x x 17 identify any points of discontinuity and determine giving reasons if 6x x 1 they are removable infinite essential or jump discontinuities 2 Points

x3 2x 2 x ax b 4x4 3x 5x 2 if 1 x find values for a and b so that the function is continuous for all x 4 Points 6 Given g x if x 2 if 2 x 1

C Average daily temperature as a function of the day of the year 1 Point

5 Find all the constants a that make the function f x 3 a x if x 3 a x if x 3 continuous for all x 2 Points

3 Suppose f x sin x2 4 1 Identify any points of discontinuity and determine giving reasons if they are removable infinite essential or jump discontinuities 3 Points

x b Given the function g x cos x that will make the function differentiable A 0 B 1 C 2 D No such value exists x 0 x 0 at x 0 Find the value of b if any E There is not enough information to determine the value

4 The domain of f x is 0 and 1 2 3 4 1 17 X f x 1 12 1 2 5 10 10 A Find an equation for f x that satisfies the domain requirements and the data table Demonstrate that your function satisfies the data points in the table Is your function even odd or neither I

Which of the following functions is continuous but not differentiable at x 1 x x 1 2x x 1 I y x 1 II y III y x x 1 1 x 1 X A I only B II only C I and III only D II and III only E All of these functions are continuous but not differentiable at x 1

Evaluate the limit if it exists Use L Hospital s rule if appropriate lim X 0 lim x 0 In cos 4x 3x In cos 4x 3x Simplify your answer

Find the derivative of the given function y 3 sin 6x dy dx 0

lim cos x x 0 X O A O OB 1 2 O C 1 OD 2 OE Does not exist

2 x x lim xoo 2x 4x4 3 O OB 1 O C 0 O D 2 O E A 4 00 c

OA OB O C 38 OD OE 7 2 4 a 2 3

lim 49 h 7 h 0 h O A 14 B 7 OC 0 OD 14 O E T

The function f x x bl intersects g x 2 x 4 at exactly one point The value of bis A 4 OB 5 C 5 D 4 E Any real number

If f x 3 then f x 3 x 3 OA 3 X OB No inverse exists O c 3 2 O D O E 3 x 3

Which of the following is always true of odd functions 1 f x f x II f xl is even III If x is even OA All of these are true B None of these are true OC I only D II only E I and III only

If g x 3x 1 and f x 9 x then which of the following will have a domain of 3 1 3 U 1 3 3 O A g f x OB f g x O C g f x OD f g x OE g f x

he domain of f x x 6x 7 O A 1 7 B 1 7 O c x 1 U x 7 D 1 U 7 E x 1 or x 7 IS

Using the function f x 5x 2x 3x x 8 and Descartes Rule of Signs which are possible combination positive negative and imaginary zeros Select all that apply 2 positive I negative 2 complex 1 1 positive 1 negative 2 complex 2 positive 1 negative 1 complex 3 positive 1 negative 0 complex 1 positive 3 negative 0 complex

Graph the following function Drag the movable red point to set the asymptotes for one period of the given function Also place the blue point at the correct set of coordinates You may click on a point to verify its coordinates Provide your answer below TT TT 2 7 6 5 4 3 2 14 0 1 2 3 4 TT 2 1 3TT 2 y 4 cot 2TT 5TT 2 3TT

y 4 cot x 1 3 Drag the movable black point to set the left vertical asymptote and shift the function the red point to set the right vertical asymptote thereby setting the period of the function and the blue point at the correct set of coordinates You may click on a point to verify its coordinates Note that the two asymptotes can be moved independently of each other and that only one period of the function is shown Provide your answer below 2TT 3TT 2 TT 4 TT 2 3 2 0 0 0 1 2 4 1 1 2 02 TT 2 3TT 2 I