Limits & Continuity Questions and Answers

Find the following limits If needed enter inf for oo and inf for a lim ln 5 11x ln 6 7x 818 b lim ln 5 sin x 1 H 0 0 4519

R log lo is the formula that gives the Richter magnitude of an earthquake where I is the intensity the earthquake and lo is the intensity of a barely felt earthquake What would the Richter magnitude be if an earthquake were 102 3 times as intense as a barely felt earthquake 1 102 30 Round your answer to the nearest tenthss and do not include the units in your answer

Graph the function not by plotting points but by starting from the graph of ye in the figure below g x 4 e 4 2 YA 0 y 10 5 10 y 3 2 y 2 y et 4 2 y 10 5 5 10 2

Graph both functions on one set of axes x 4 and g x O f 8 2 y 20 r 10 10 20 20 10 2 X 4 2 f g 20 10 10 20 20 y 10 2 f 8

The natural logarithmic function f x In x has th Select asymptote x horizontal vertical

Perform the elementary row operatiol 1 2 4 0 4 2

9 22 Evaluating Logarithms evaluate the expression 9 log 50 log 200 11 log 60 log 15 13 log3 81

Evaluate the following limits COS os x 1 x4 14x8 lim x 0 lim H 0 cos x 1 2 1428 11 11

Find the following limits If needed enter inf for co and inf for 0 a lim ln 5 11x ln 6 7x b lim ln 5 sin x

Evaluate the indicated limit lim 1 372 818 7x I Hint One effective approach uses the power series for In 1 z valid

alth Sciences Tut pard My courses SC MATH1507 M ne Assignments Assignment 4

angies and Finding Angle K For the figure shown

Solve the system of equations graphically If the syste a solution state whether the system is inconsistent or

4 This problem concerns a function about which the following information is known f is a differentiable function defined at every real number a f 0 1 2 y f x has its graph given at center in Figure 3 1 17 24 A 1 Figure 3 1 17 At center a graph of y f r at left axes for plotting y f x at right axes for plotting y f x a Construct a first derivative sign chart for f Clearly identify all critical numbers of f where f is increasing and decreasing and where f has local extrema b On the right hand axes sketch an approximate graph of y f x c Construct a second derivative sign chart for f Clearly identify where fis concave up and concave down as well as all inflection points d On the left hand axes sketch a possible graph of y f x

a Factor f x x 5x 81x 405 into factors of the form x c given that 5 is a zero b Solve x 5x 81x 405 0 Part 0 2 Part 1 of 2 a Factor f x x 5x 81x 405 into factors of the form x c given that 5 is a zero x 5x 81x 405 i

a Graph f x 6 and g x 7 on the same Cartesian plane and shade the region bounded by the y axis f x 6 and g x 7 b Solve f x g x a Choose the graph below that shows the graphs of f x 6 and g x 7 with the region bounded by the y axis f x 6 and g x 7 shaded The window display is 5 5 1 by 1 14 1 O B OA O C D b At x f x g x Type an integer or a decimal rounded to three decimal places as needed TTTTTT

Find the derivative of the functio y 14x 6x4 y

If a is a Quadrant IV angle with cos a find a sin a B b cos a B 15 15 4 and sin B 16 16 where B T 2

Find the exact value of the expression below COS s 730 cos 13 sin 73 sin 13

Rewrite sin 3 2 37 4 in terms of sin x and cos x

If sin t sint cost A tan t A tant 4 then

In the Theory of Relativity the length of an object depends on how fast it is moving its velocity v with respect to an observer If an object is 18 meters long when it is at rest the function L v which gives its length is L v 18 1 where c is the speed of light Calculate the length of this object for velocities which are the following percentages of c L 0 3c L 0 5c L 0 7c v c meters meters meters

Convert the angle 120 to radians Give the exact value and use pi for 2 09 x no decimals allowed

the system of equations using Cramer s Rule 8x 48 x 8y 0

Uninhibited growth can be modeled by exponential functions other than A t of the population at time t An insect population grows exponentially Comple

Prove or disprove the following Make sure to explain every step in detail and use plenty of ords to indicate what follows from what logically 4 x y X Y a For any positive real numbers x and y we have b For any integer n if n n 15 0 then 2n 2n 3 0 c If x is a real number such that x 3 5 then x 1

2 MORE NUMBER SYSTEMS SETS AND QUANTIFIERS Rewrite the following as formulas with quantifiers a There is a natural number which is not the sum of three square integers b For any positive real number there is a smaller positive rational number c If the product of two integers is an integer multiple of 3 then at least one of them is an integer multiple of 3

1 Solve the triangle if possible Round answers to the nearest tenth Include a sketch of the triangle labeled with the given information B 22 b 5 and a 24 2 Solve the triangle if possible Round answers to the nearest tenth Include a sketch of the triangle labeled with the given information A 42 b 2 4 and a 2 1

c Determine the exact value for each of the following expressions 3 pts each arcsin a b tan sec arctan 4 sin arccos of the following triangles 9 pis each

Solve the system of equations graphic ordered pair as a solution state wheth 1 y x 1

Draw the graph of the function and state the domain and

of 23 b b 3y 2 ob 1 3 b Nie 2 Use it to s

Determine the interval or number over which the function will be continuous f x x 2 x 6 x 2

Solve for z in the equa Note that z 0 log 16 2

Suppose f is a continuous function where f 9 6 and f 9 10 The equation of the line tangent to the graph of y f x at x 9 will be y So f x Based on this f 9 6 A solution to f x 0 is x for a close to x 9

a Find the linear approximation for the following function at the given point b Use part a to estimate the given function value f x y 6x 9y 6xy 5 6 estimate f 4 9 5 97 a L x y

A bridge is to be built across a small lake from a gazebo to a dock The bearing from the gazebo to the dock is 5 41 W From a tree x 110 meters from the gazebo the bearings to the gazebo and the dock are S 74 E and S 28 E respectively see figure Find the distance from the gazebo to the dock Round your answer to the nearest whole number m Tree Dock xm Gazebo

REE RESPONSE nd the exact value of the given expression be Must show your work and draw a right triangl tan cos 3 721

6 5 Amplitude Period 12 Midline y 4 4 Based on the graph above determine the amplitude midline and period of the function 8 9 q

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 f x 25 2 Al 4 3 2 1 2 54 1 2 3 The curve above is the graph of a sinusoidal function It goes through the points 8 0 and 6 0 Find a sinusoidal function that matches the given graph If needed you can enter T 3 1416 as pi in your answer otherwise use at least 3 decimal digits

Find a function of the form y Asin kx C or y A cos kx C whose graph matches this one 13 12 11 10 9 8 y 6 5 4 3 2 5 1 3 2 1 1 2 3 Leave your answer in exact form if necessary type pi for 3 4 5 6

g x 32x and f x x 1 Solve for x such that g x f g x

4pts Fin d L is t E xplain the Leading Term End Behavior Number of Zeros Multiplicity of the Zeros the Max Number of Turning Points and the y intercept for P x x 2 x 1 2 x 3

Use the Laws of Logarithms to expand the expression In ab

ning Time 1 hour 50 minutes 35 seconds on Completion Status we Tor um prvem me hs of two functions y f x in blue and y g x in red are shown Explain with giving the reason whether the limit given below exists or does not exist 34 553 3 y f x 2 1 DABU Z 3 1 lim f x 8 x 2 stv 2 3

Explain with giving the reason whether lim f x exists or does not exist f x x 3 9 x X 1 if x 3 if x 3

Suggested time for this proble 1 Complete the table below Round your answers to the nearest ten thousandth 2 Based on your table guess the value of lim x x The limit is equal to x 0 3 Check that your guess is correct by checking whether it possible to make In x 1 the limit you guessed in Part 2 as small as we wish For example to make In x 1 the limit you guessed in Part 2 0 00001 it suffices to take 0 x x 0 2 0 1 0 01 0 001 0 0001 In x 1

Find the value of the constant c so the function f is continuous on 0 whe f a S ca 5 if a 8 c a 5 if a 8

Let f x 3x 8 182x if if x 2 2 Determine whether f x is continuous at a 2 If f x is not continuous identify why Not continuous f 2 is undefined Not continuous lim 2 f x f a f x does not exist Not continuous lim 2 The function is continuous at 2

Given the function below determine whether the function is continuous at the point x 1 If not indicate why f x 2x 6 I 1 4 x 1 2x 2 I 1 Continuous at x 1 Not continuous f 1 does not exist Not continuous lim f x does not exist 2 1 Not continuous f 1 and limit exist but are not equal