Limits & Continuity Questions and Answers

1 Points DETAILS LARLINALG8 7 3 047 0 1 Submissions Used ASK YOUR TEACHER Find a matrix P such that PTAP orthogonally diagonalizes A Verify that PTAP gives the proper diagonal form Enter each matrix in the form row 1 row 2 where each row is a comma separated list A P PTAP 0 10 10 10 50 10 0 5 2 1 1 1 1 1 2 2 1 0 0 0 0 15 0 0 0 15 MY NOTES Show My Work Optional

11x 4 OA 9x 4 3x OB 4 OC 2n 5x OD A given angle

ind the domain of the function x log 10 x 13x 42 OA 0 6 U 7 00 OB 7 00 OC 6 7 OD 6

For f x in the figure below find the x values of the global maximum and global minimum on the domain 8 x 4 30 20 10 12 8 J f x 4 8 12 16 Enter your answers to the nearest integer The global maximum occurs when x The global minimum occurs when x i

Use the Laws of Logarithms to expand the expression In 55 8s

9 Evaluate lim h 0 tan TT 4 h 1 derivative

1 Find the limit Use l Hospital s Rule where appropriate If I Hospital s Rule does not apply explain why x a b lim x 01 cos x In x lim x 0 X 4118 lim xln xlm 1 1

angle that satisties the given equation Express your answer as an inverse trigonometric function and as the measure of 0 in degrees 1 sin 0 2 Express 0 as an inverse trigonometric function 0 Type an exact answer using radicals as needed Use integers or fractions for any numbers in the expression

Find the limits in a through c below for the function f x OA lim f x Simplify your answer X 6 A lim f x Simplify your answer X 6 OB The limit does not exist and is neither b Select the correct choice below and fill in any answer boxes in your choice A lim f x x 3x 18 X 6 Simplify your answer o nor o Use co and co when appropriate OB The limit does not exist and is neither co nor o c Select the correct choice below and fill in any answer boxes in your choice

The total sales of a company in millions of dollars t months from now are given by the following formu S t 7 t 1 A Use the four step process to find S t B Find S 15 and S 15 C Use the results in part B to estimate the total sales after 16 months and 17 months A S t B S 15 Type an integer or a decimal S 15 Type an integer or a decimal C S 16 Type an integer or a decimal S 17 Type an integer or a decimal

Use the graph of the function f shown to estimate the indicated limits and the function value Complete parts A through D OA f 4 Type an integer 4 B The function is not defined at x 4 Af x 4 2 D Find the function value f 4 Select the correct choice below and if necessary fill in the answer box to complete your choice G

Refer to the graph of y f x to the right to describe the behavior of lim f x Use co and co where appropriate x 1 Select the correct choice below and fill in any answer boxes in your choice OA lim f x x 1 OB The limit does not exist and is neither co nor co 10 8 6 Ay 10 10

Find the indicated derivative P d du 9u04 6u 4 te balone Marilia Hant

Write an equation of the line containing the given point and parallel to the given line Express your answer in the form y mx b 2 6 5x 9y 2 The equation of the line is y Simplify your answer Use integers or fractions for any numbers in the expression

Suppose f x 14 H a The rectangles in the graph on the left illustrate a left endpoint Riemann sum for f x on the interval 4 x 6 The value of this left endpoint Riemann sum is and it is an the area of the region enclosed by y f z the x axis and the vertical lines 4 and 6 b The rectangles in the graph on the right illustrate a right endpoint Riemann sum for f x on the interval 4 6 The value of this right endpoint Riemann sum is and it is an the area of the region enclosed by y f x the x axis and the vertical lines 4 and 6 Left endpoint Riemann sum Right endpoint Riemann sum

Estimate the areas under the curve by computing the left Riemann sum L2 F1 16 1 8 1 0

Given the function f 2 compute the left endpoint Riemann sum using m 6 on the interval 1 11 Ls help numbers

Use the figure and trigonometric identities to find the exact value of the trigonometric function below in two ways a 4 cos a sin B 4 cos a sin B 4 cos a sin B 4 V 12 2 2 B

Refer to functions p and q Evaluate q p x and write the domain in interval notation Write your answers as integers or simplified fractions p x x 3x Part 0 2 Part 1 of 2 q p x 1 q x 2 x 2 0 0

Apply the Ratio Test to determine convergence or divergence or state that the Ratio Test is inconclusive n 2n p lim Enter inf for o 71 00 00 n 1 n 2n is A convergent B divergent C The Ratio Test is inconclusive n 1

X Given f x 3x 2x 1 What is the degree of the function Good job 4

5 10 TT 1 5 VE 20 15 10 5 0 5 10 is 5 10 tion 5 Use the graph to determine the denominator of the rational function Write the denominator as a factor Submit

Find where x 2x 0 Write your answer in interval notation Round to one decimal place as needed Left Interval 8 Dight Interval Submit

Find where x4 2x 0 2 Write your answer in interval notation Round to one decimal place as needed Left Interval

f 3500 is invested at an interest rate of 5 25 per year compounded continuously find the value of the investment after the given number of years Round your answers to the nearest cent a 2 years b 4 years c 6 years

The function f x ex is called the natural Need Help Read It exponential function The number e is approximately equal to Round your answer to five decimal places

Describe the long run behavior of f x 5 1 3 As a f x O As I f x Question Help Video Message instructor

Use a calculator to solve the following equation in the interval 0 2 correct to five decimal places 4 sin x 2 0 0 78540 2 35619 3 92699 5 49779 0 78540 0 78540 2 35619 3 92699 0 78540 2 35619 3 92699 5 49779 7 06858 0 78540 2 35619

Use an addition or subtraction formula to find the exact value of the expression tan 255 0 1 3 O2 3 2 01 1 3 1 V 3

Find all solutions of the equation 2 sin x 1 0 Select the correct answer where k is any integer TU TU 6 TU 6 TU 6 2k KI 5TT 6 5TT Km KT 6 2k 2k 11 6 11 KT 2k

Find the horizontal asymptote of f x y 4x x 4 3x x 2

3 2 f x x 4x 4x Find all real numbers a so that the function dy continious at a X

lim X sin x X COS X X

2 The parent function broken line is y f x Describe the exact transformations to its image then write the coordinate rule Write the equation of its image in function form Description Coordinate Rule Parent y f x Image g x 7 6 5 A 34 2 1 2 3 y 2 3 16

Use transformations of the graph of f x to identify the graph of the function given below Use the graph to determine its domain range and asymptote g x 1 Identify the graph of g x e 1 OA What is the domain of g x 12 Use interval notation What is the range of g x e 1 Use interval notation What line is the horizontal asymptote of g x 1 OB SITE

a Replace the polar equation 8rsine with the equivalent Cartesian equation t

In Exercises 13 22 find the limit of each rational function a as x and b as x 13 f x 2x 3 x 12 547 x 1 15 f x 2 17 h x 19 g x 21 h x x3 3 7x 3x 6x 10x5 x 31 6 2r 3r 2x 3 5x 3x

10 5 10 LO Af x 15 X 10 C

Find and simplify the expression if f x x 3 f x 2 f x 2 Cimplic

7 2 9 90 16 1 3 112X 7

Recall that for 1 x 1 the Taylor expansion for f x a lim 1 1 c which is also known as a geometric series Use this result to calculate the following limits b 11 400 1 1 1 3 3 3 3 4 8 3 16 32 64 9 27 81 1 1 z 1 3 n centred at z 0 is given by 1 x x x x Note Both of these objects are generally referred to as infinite sums or series The first expression explicitly gives the nth term and asks you to calculate the limit as noo whereas the second implicitly suggests via the that what you are calculating is an infinite sum of terms Althoug they look different they are asking for the same thing for you to calculate the sum of all their infinite terms

Suppose that the second order Taylor polynomial of some function f about x 0 is given by x x 7 24 How consider the function 5 x 3 T x 3 Find S x the second order Taylor Polynomial for g about x 0 g x f x e x 2347 168

Follow the steps for graphing a rational function to graph the function R x ww 3x 3 5x 25 Simplify your answer Type an integer or a fraction OC The graph has x intercept s and no y intercept Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once OD The graph has neither x intercepts nor y intercepts Determine the behavior of the graph of R at any x intercepts Select the correct choice below and if necessary fill in answer box es to complete your choice O A The graph will cross the x axis at x and touch but not cross the x axis at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OB The graph will cross the x axis at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer once x intercent OC The graph will touch but not cross the x axis at x Type an integer or simplified fraction Use a comma to separate answers as needed Type each answer of once

f x 1 sin if x 0 if x 0 0 a Prove that 1 f x 1 for all x b Determine the value of lim o f x if it exists If it does not e explain why

j limx 2 2 x2 421 8 m lim 0 cos r 1 x Hin k lim 4 4 n lim 2 32 10 x 4x 4 www 3r 2r 5 1 limr 4 6 r 3x 2 o lim 1 2 3 x

3 Consider the function k x r sin 1 Show that 0 k r x for a x 0 Find lim o k r using the Squeeze Theorem

e lim o tan x I 2x 3x 1 f lim 1x 37 2r 5

A company s total cost in millions of dollars is given by C t 130 60e where t is the time in years since the start up date The graph of C t is shown to the right Find each of the following a The marginal cost C t b C 0 c C 3 d Find lim C t and lim C t 10 t Do not include the symbol in your answer a C t b C 0 million per year Simplify your answer Do not include the symbol in your answer million d lim C t 1 Simplify your answer Do not include the symbol in your answer lim C t 118 per year wor Do not include the symbol in your answer Ay 150 75 0 c C 3 per year Simplify your answer Round to the nearest thousand as needed Do not include the symbol in your answer 0 15 30 Q Q

Solve the inequality Enter your answer using interval notation 9x 49 x 25 0

Solve the inequality Enter your answer using interval notation x 4x 5 5x222x 15 0