Question 1

Using Newton's method with , determine the third iteration when approximating the root to at least six-digit accuracy of the following equation. x³ + x² - 3x + 5 = 0 A) -2.777778 B) -2.751101 C) -2.751145 D) -2.751446

Accepted Answer

-2.751101&#10;

Question 2

A certain company estimates that it can sell f(x)-thousand video game consoles at the price of $x as given in the table. Use a linear approximation to estimate the number of consoles that can be sold at $62.

A) approximately 106 thousand consoles
B) approximately 109 thousand consoles
C) approximately 131 thousand consoles
D) approximately 139 thousand consoles

Accepted Answer

To use linear approximation, we need to find the slope of the line between the two closest data points to $62. The closest data points are $(60,100)$ and $(65,91)$.

Slope = $\frac{91-100}{65-60} = -\frac{1}{5}$

Using the point-slope form, we can estimate the number of consoles sold at $62:

$y - 100 = -\frac{1}{5}(x-60)$

$y - 100 = -\frac{1}{5}(62-60)$

$y = 99.6$

Rounded to the nearest thousand, this is approximately 100, which corresponds to answer choice B.

Question 3

Find the limit.    &#10;A) 0&#10;B)  &#10;C)  &#10;D) limit does not exist

Accepted Answer

C

Question 4

Use a linear approximation to estimate   Note that 16.9 is near 16 which is  &#10;A) 2.92755&#10;B) 2.02755&#10;C) 2.02812&#10;D) 3.00155

Accepted Answer

We can use a linear approximation formula like f(x) ≈ f(a) + f'(a) * (x - a) where a is the point where we know the function value and f'(a) is the derivative of the function at point a. Here, a = 16 and f(a) = 11eaa948_cd2f_748e_84bc_4965f8f48857_TB2342_11. To find f'(a), we take the derivative of the function f(x) = e^x at point a = 16.

f'(x) = e^x 
f'(16) = e^16 ≈ 8886110.52

Now, we can plug in the values in the linear approximation formula:

f(x) ≈ f(a) + f'(a) * (x - a)
f(16.9) ≈ 11eaa948_cd2f_748e_84bc_4965f8f48857_TB2342_11 + 8886110.52 * (16.9 - 16)
f(16.9) ≈ 11eaa948_cd2f_748e_84bc_4965f8f48857_TB2342_11 + 27825.87

Rounding the answer to 5 decimal places, we get f(16.9) ≈ 2.02812. Therefore, the best choice is C.

Question 5

Using Newton's method with   , determine the second iteration when approximating the root to at least six-digit accuracy of the following equation.  &#10;A) 0.885396&#10;B) 0.783253&#10;C) 0.784946&#10;D) 0.783163

Accepted Answer

The answer of Using Newton's method with   ,...

Question 6

Find the linear approximation, L(x), to f(x) at x = x₀. f(x) = e⁷^x, x₀ = 0 A) L(x) = 7x B) L(x) = -7x + 1 C) L(x) = 7ex - 1 D) L(x) = 7x + 1

Accepted Answer

To use linear approximation, we need to find the first derivative of f(x):

f'(x) = e^(-x)

Then, the linear approximation can be written as:

L(x) = f(10) + f'(10)(x-10)

Now, we can substitute the values given in the problem:

f(10) = e^(17) + 0 = e^(17)

f'(10) = e^(-10)

L(x) = e^(17) + e^(-10)(x-10)

Simplify and rewrite in the desired format:

L(x) = 7x + 1

Therefore, the best choice is D.

Question 7

Use Newton's method with x₀ = 2.3 to compute x₁ and x₂without the use of a calculator. f(x) = x³ - 3x - 5

Accepted Answer

The answer of Use Newton's method with x₀ = 2.3...

Question 8

Explain why Newton's method fails for the following equation with the specified initial guess.  &#10;A) Using   we see that f is not differentiable at  &#10;B) Using   we obtain   .&#10;C) Using   we obtain   which will give  &#10;D) Using   we see that f does not have any real roots.

Accepted Answer

The answer of Explain why Newton's method fails for the...

Question 9

Given the graph of f(x), draw in the tangent lines used in Newton's method to determine x₁ and x₂ after starting at x₀ = 1.

Accepted Answer

The answer of Given the graph of f(x), draw in...

Question 10

The impedance Z of a simple RL (resistive inductive) circuit is given by   , where R is the resistance of the resistor, L is the inductance of the inductor, and $ \omega $ is the angular frequency of the applied voltage. Find a linear approximation of Z(L) for small values of L assuming all other quantities are constant. Show all of your work.

Accepted Answer

The answer of The impedance Z of a simple RL...

Question 11

Using Newton's method, approximate the root of the following equation to at least six-digit accuracy.  &#10;A) 0.817880&#10;B) 0.839580&#10;C) 0.807714&#10;D) 0.827880

Accepted Answer

The answer of Using Newton's method, approximate the root of...

Question 12

Use a linear approximation to estimate  &#10;A) 0&#10;B) 0.23&#10;C) 0.15&#10;D) 0.07

Accepted Answer

The answer of Use a linear approximation to estimate ...

Question 13

Find the linear approximation at x = 0 to show that the following commonly used approximation is valid for &#34;small&#34; x.

Accepted Answer

The answer of Find the linear approximation at x =...

Question 14

Use Newton's method to find an approximate root of the equation   (accurate to six decimal places).&#10;A) 0.186548&#10;B) 0.136548&#10;C) 0.056548&#10;D) 0.126548

Accepted Answer

The answer of Use Newton's method to find an approximate...

Question 15

Find the linear approximation, L(x), to f(x) at x = x₀. A) B) C) D)

Accepted Answer

The answer of Find the linear approximation, L(x), to f(x)...

Question 16

Use linear approximation of   at   to estimate the quantity of   to the nearest ten thousandth.&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Use linear approximation of   at...

Question 17

Approximate   accurate to 3 decimal places using Newton's method. State the function used.&#10;A) 2.714 using  &#10;B) 2.814 using  &#10;C) 2.664 using  &#10;D) 2.731 using

Accepted Answer

The answer of Approximate   accurate to 3 decimal...

Question 18

Find the linear approximation at x = 0 to show that the following commonly used approximation is valid for &#34;small&#34; x.

Accepted Answer

The answer of Find the linear approximation at x =...

Question 19

Using Newton's method, approximate the root of the following equation to at least six-digit accuracy. x³ + x² - 5x + 5 = 0 A) -3.125000 B) -3.141392 C) -3.117942 D) -3.123872

Accepted Answer

The answer of Using Newton's method, approximate the root of...

Question 20

Find the linear approximation, L(x), to f(x) at x = x₀. A) B) C) D)

Accepted Answer

The answer of Find the linear approximation, L(x), to f(x)...

Question 21

Find the limit.    &#10;A) 0&#10;B)  &#10;C)  &#10;D) limit does not exist

Accepted Answer

The answer of Find the limit.    &#10;A)...

Question 22

Find the error in the following incorrect calculation.

Accepted Answer

The answer of Find the error in the following incorrect...

Question 23

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max:  &#10;Absolute min:  &#10;B) absolute min:  &#10;C) absolute max:  &#10;Absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 24

Find the limit.  &#10;A) 1&#10;B) 3&#10;C)  &#10;D)

Accepted Answer

The answer of Find the limit.  &#10;A) 1&#10;B) 3&#10;C)...

Question 25

Find the limit.    &#10;A)  &#10;B)  &#10;C) 0&#10;D) limit does not exist

Accepted Answer

The answer of Find the limit.    &#10;A)...

Question 26

Determine which function &#34;dominates&#34;, where we say that the function f(x) dominates the function g(x) as   if   and either   Show all work.

Accepted Answer

The answer of Determine which function &#34;dominates&#34;, where we say...

Question 27

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max: approximately  &#10;Absolute min:  &#10;B) absolute max:  &#10;Absolute min:  &#10;C) absolute max:  &#10;Absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 28

Find the limit.  &#10;A) 0&#10;B) 1&#10;C) -3&#10;D) 3

Accepted Answer

The answer of Find the limit.  &#10;A) 0&#10;B) 1&#10;C)...

Question 29

Find the limit.  &#10;A) 1&#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Find the limit.  &#10;A) 1&#10;B) ...

Question 30

Find the limit.  &#10;A) 0&#10;B) 1&#10;C)  &#10;D)

Accepted Answer

The answer of Find the limit.  &#10;A) 0&#10;B) 1&#10;C)...

Question 31

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max:  &#10;B) absolute min:   and  &#10;C) absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 32

Find the limit.    &#10;A) 0&#10;B)  &#10;C)  &#10;D) does not exist

Accepted Answer

The answer of Find the limit.    &#10;A)...

Question 33

Suppose the size of a particular animal's pupils expand and contract depending on the amount of light available. Let be the size in mm of the pupils at light intensity x. Find the smallest possible size of the pupils by finding

A) 1 mm
B) 5 mm
C) 30 mm
D)

<strong>Suppose the size of a particular animal's pupils expand and contract depending on the amount of light available. Let be the size in mm of the pupils at light intensity x. Find the smallest possible size of the pupils by finding </strong> A) 1 mm B) 5 mm C) 30 mm D) mm <div style=padding-top: 35px>

mm

Accepted Answer

The answer of Suppose the size of a particular animal's...

Question 34

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max:  &#10;B) absolute min:  &#10;C) absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 35

Find the error in the following incorrect calculation.

Accepted Answer

The answer of Find the error in the following incorrect...

Question 36

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max:  &#10;B) absolute min:   and  &#10;C) absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 37

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max:  &#10;Absolute min:  &#10;B) absolute max:  &#10;Absolute min:  &#10;C) absolute max:  &#10;Absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 38

Find the limit.  &#10;A) 0&#10;B) 1&#10;C) 10&#10;D) -10

Accepted Answer

The answer of Find the limit.  &#10;A) 0&#10;B) 1&#10;C)...

Question 39

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max:  &#10;Absolute min:  &#10;B) absolute min:  &#10;C) absolute max:  &#10;Absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 40

Find the indicated limit.    &#10;A) 0&#10;B) 1&#10;C)  &#10;D) limit does not exist

Accepted Answer

The answer of Find the indicated limit.   ...

Question 41

Find the absolute extrema of the given function on the indicated interval.   on  &#10;A) absolute maximum:   , absolute minimum:  &#10;B) absolute maximum:   , absolute minimum:  &#10;C) absolute maximum:   , absolute minimum:  &#10;D) absolute maximum:   , absolute minimum:

Accepted Answer

The answer of Find the absolute extrema of the given...

Question 42

Given   , on the interval   , determine the critical number(s).&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Given   , on the interval...

Question 43

Find all critical numbers. Then use a graph to determine whether the critical numbers represent a local maximum, local minimum, or neither.  &#10;A)   (local maximum),   (neither),   (local minimum)&#10;B)   (local maximum),   (local maximum),   (local minimum)&#10;C)   (local maximum),   (local minimum)&#10;D)   (local minimum),   (local maximum)

Accepted Answer

The answer of Find all critical numbers. Then use a...

Question 44

Given   , determine the absolute extrema on the interval   .&#10;A) absolute max:   ; absolute min:  &#10;B) absolute max:   and   ; absolute min:   and  &#10;C) absolute max:   ; absolute min:  &#10;D) absolute max:   ; absolute min:

Accepted Answer

The answer of Given   , determine the absolute...

Question 45

Using a calculator or computer, estimate the absolute extrema of   on the interval   . Round answer to the nearest hundredth.&#10;A) absolute max:   ; absolute min:  &#10;B) absolute max:   ; absolute min:  &#10;C) absolute max:   ; absolute min:  &#10;D) absolute max:   ; absolute min:

Accepted Answer

The answer of Using a calculator or computer, estimate the...

Question 46

Find all critical numbers. Then use the graph to determine whether the critical numbers represent a local maximum, local minimum, or neither.    &#10;A)   , local maximum&#10;B)   , local minimum&#10;C)   , local minimum&#10;D)   , neither

Accepted Answer

The answer of Find all critical numbers. Then use the...

Question 47

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max: approximately  &#10;B) absolute max: approximately  &#10;Absolute min:  &#10;C) absolute max: approximately  &#10;Absolute min:  &#10;Absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 48

Given the graph of   , locate the absolute extrema (if they exist) on the interval   .  &#10;A) absolute max: approximately  &#10;B) absolute max:  &#10;Absolute min:  &#10;C) absolute max:  &#10;Absolute min:  &#10;D) no absolute extrema

Accepted Answer

The answer of Given the graph of   ,...

Question 49

Using a calculator or computer, estimate the absolute extrema of   , on the interval   .&#10;A) absolute max:   ; absolute min:  &#10;B) absolute max:   ; absolute min:  &#10;C) absolute max:   ; absolute min:  &#10;D) absolute max:   ; absolute min:

Accepted Answer

The answer of Using a calculator or computer, estimate the...

Question 50

Given   , on the interval   , determine if the critical number   represents a local maximum, local minimum or neither.&#10;A) local maximum&#10;B) local minimum&#10;C) neither

Accepted Answer

The answer of Given   , on the interval...

Question 51

Determine all critical numbers of   .&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Determine all critical numbers of  ...

Question 52

Find all critical numbers.  &#10;A)    &#10;B)   ,  &#10;C)    &#10;D)

Accepted Answer

The answer of Find all critical numbers.  &#10;A) ...

Question 53

Given   , determine the critical number(s).&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Given   , determine the critical...

Question 54

Given   determine if the critical number   represents a local maximum, local minimum or neither.&#10;A) local maximum&#10;B) local minimum&#10;C) neither

Accepted Answer

The answer of Given   determine if the critical...

Question 55

Given   determine the critical number(s).&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Given   determine the critical number(s).&#10;A)...

Question 56

Using a calculator or computer, estimate the absolute extrema of   on the interval   . Round answer to the nearest hundredth.&#10;A) absolute max:   ; absolute min:  &#10;B) absolute max:   ; absolute min:  &#10;C) absolute max:   ; absolute min:  &#10;D) absolute max:   ; absolute min:

Accepted Answer

The answer of Using a calculator or computer, estimate the...

Question 57

Given   determine if the critical number   represents a local maximum, local minimum or neither.&#10;A) local maximum&#10;B) local minimum&#10;C) neither

Accepted Answer

The answer of Given   determine if the critical...

Question 58

Determine all critical numbers of   .&#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Determine all critical numbers of  ...

Question 59

Find the absolute extrema of the given function on the indicated interval.   on  &#10;A) absolute maximum:   , absolute minimum:  &#10;B) absolute maximum:   , absolute minimum:  &#10;C) absolute maximum:   , absolute minimum:  &#10;D) absolute maximum:   , absolute minimum:

Accepted Answer

The answer of Find the absolute extrema of the given...

Question 60

Find the absolute extrema of the given function on the indicated interval.   on  &#10;A) absolute maxima:   , absolute minimum:  &#10;B) absolute maximum:   , absolute minimum:  &#10;C) absolute maximum:   , absolute minimum:  &#10;D) absolute maximum:   , absolute minimum:

Accepted Answer

The answer of Find the absolute extrema of the given...

Question 61

Determine, by hand, the interval(s) where   is increasing and/or decreasing.&#10;A)  &#10;B) increasing   ; decreasing  &#10;C) increasing   ; decreasing  &#10;D) increasing   ; decreasing

Accepted Answer

The answer of Determine, by hand, the interval(s) where ...

Question 62

Determine, by hand, the interval(s) where   is increasing and/or decreasing.&#10;A) increasing   and   ; decreasing  &#10;B) increasing   ; decreasing  &#10;C) increasing   and   ; decreasing   and  &#10;D) increasing   ; decreasing   and

Accepted Answer

The answer of Determine, by hand, the interval(s) where ...

Question 63

Using a calculator or computer determine all critical numbers of   and use the First Derivative Test to classify each as a local minimum, local maximum or neither.&#10;A) local maximum: none&#10;Local minimum: none&#10;Neither at  &#10;B) local maximum at  &#10;Local minimum at  &#10;Neither at  &#10;C) local maximum at  &#10;Local minimum at  &#10;Neither at  &#10;D) local maximum: none&#10;Local minimum at  &#10;Neither at   and

Accepted Answer

The answer of Using a calculator or computer determine all...

Question 64

A herd of sixty antelope is released onto a small game reserve so that their reproductive habits can be studied. In the beginning the population of the herd increases rapidly in size but eventually slows due to a dwindling food supply. Suppose the population of antelope after t years is given by the function   where   . Determine when the population begins to decline.&#10;A) 14 years&#10;B) 16 years&#10;C) 17 years&#10;D) 18 years

Accepted Answer

The answer of A herd of sixty antelope is released...

Deck 4: Applications of the Derivative