if 2 pounds of strawberries cost 4.50 how much would 3 pounds cost
Find the second degree Taylor polynomial for f(x)= sqrt(x^2+8) at the number x=1
Answer:
[tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightAlgebra I
Functions
Function NotationCalculus
Differentiation
DerivativesDerivative NotationDerivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Taylor Polynomials
Approximating Transcendental and Elementary Functions[tex]\displaystyle P_n(x) = \frac{f(c)}{0!} + \frac{f'(c)}{1!}(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \frac{f'''(c)}{3!}(x - c)^3 + ... + \frac{f^n(c)}{n!}(x - c)^n[/tex]Step-by-step explanation:
*Note: I will not be showing the work for derivatives as it is relatively straightforward. If you request for me to show that portion, please leave a comment so I can add it. I will also not show work for elementary calculations.
Step 1: Define
Identify
f(x) = √(x² + 8)
Center: x = 1
n = 2
Step 2: Differentiate
[Function] 1st Derivative: [tex]\displaystyle f'(x) = \frac{x}{\sqrt{x^2 + 8}}[/tex][Function] 2nd Derivative: [tex]\displaystyle f''(x) = \frac{8}{(x^2 + 8)^\bigg{\frac{3}{2}}}[/tex]Step 3: Evaluate
Substitute in center x [Function]: [tex]\displaystyle f(1) = \sqrt{1^2 + 8}[/tex]Simplify: [tex]\displaystyle f(1) = 3[/tex]Substitute in center x [1st Derivative]: [tex]\displaystyle f'(1) = \frac{1}{\sqrt{1^2 + 8}}[/tex]Simplify: [tex]\displaystyle f'(1) = \frac{1}{3}[/tex]Substitute in center x [2nd Derivative]: [tex]\displaystyle f''(1) = \frac{8}{(1^2 + 8)^\bigg{\frac{3}{2}}}[/tex]Simplify: [tex]\displaystyle f''(1) = \frac{8}{27}[/tex]Step 4: Write Taylor Polynomial
Substitute in derivative function values [Taylor Polynomial]: [tex]\displaystyle P_2(x) = \frac{3}{0!} + \frac{\frac{1}{3}}{1!}(x - c) + \frac{\frac{8}{27}}{2!}(x - c)^2[/tex]Simplify: [tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - c) + \frac{4}{27}(x - c)^2[/tex]Substitute in center c: [tex]\displaystyle P_2(x) = 3 + \frac{1}{3}(x - 1) + \frac{4}{27}(x - 1)^2[/tex]Topic: AP Calculus BC (Calculus I + II)
Unit: Taylor Polynomials and Approximations
Book: College Calculus 10e
The second degree Taylor polynomial for the function [tex]f(x) = sqrt(x^2+8)[/tex] at x=1 is [tex]T(x) = 3 + (x-1) - 1/32(x-1)^2[/tex].
To find the second degree Taylor polynomial for [tex]f(x) = sqrt(x^2+8)[/tex] at the number x=1, we begin by calculating the necessary derivatives and evaluating them at x=1. The Taylor polynomial of degree n at x=a is given by:
[tex]T(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n[/tex].
In this case, we need to find the first and second derivatives:
[tex]f'(x) = \frac{1}{2}(x^2+8)^{-1/2} · 2x[/tex]
[tex]f''(x) = \frac{1}{2} · (-1/2)(x^2+8)^{-3/2} · 2x^2 + \frac{1}{2}(x^2+8)^{-1/2}[/tex]
Then we evaluate f(x), f'(x), and f''(x) at x=1:
[tex]f(1) = sqrt(1^2+8) = sqrt9 = 3[/tex]
[tex]f'(1) = \frac{1}{2}(1^2+8)^{-1/2} · 2 · 1 = 1[/tex]
[tex]f''(1) = \frac{1}{2} · (-1/2)(1^2+8)^{-3/2} · 2 · 1^2 + \frac{1}{2}(1^2+8)^{-1/2} = -\frac{1}{16}[/tex]
Thus, the second degree Taylor polynomial at x=1 is:
[tex]T(x) = 3 + (x-1) - \frac{1}{32}(x-1)^2[/tex].
For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?
The equation of the parabola is y = ax² = (-1/2)x² where a = -2 and b = -8 are the required values of a and b.
What is the slope of the tangent line?The first derivative of y = ax² that represents the slope of the tangent line to the curve of y = ax² .
Here, dy/dx = 2ax.
When x = 2,
dy/dx = 2a(2) = 4a.
The point of tangency is (2,y), where y = a(2)², or y=4a;
thus, the point of tangency is; (2, 4a).
The equation of the tangent line to y=ax² at (2,4a)
Now differentiating y=ax² with respect to x,
dy/dx = 2ax
dy/dx (at 2,4a) = 2a(2) = 4a
The line 2x + y = b is tangent to y = ax² at (2,4a).
The slope of 2x + y = b can be found by solving 2x + y = b for y:
y = b - 2x
Slope m = -2
Thus, dy/dx = 4a = - 2, and thus a = -2/4, or a = -1/2. All we have to do now is to find the value of b.
We know that 2x + y = b, then if x=-2 and y=-8,
2(-2) + [-8] = b = -4 - 8 = -12
Thus, the equation of the parabola is y = ax² = (-1/2)x² where a = -2 and b = -8 are the required values of a and b.
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Solve for the distance between (522, 1322) and (9000, -1337) to the third decimal.
MARIA WALKED 4.035 KILOMETERES. WHAT IS 4.035 WRITTEN IN AS EXPANDED FORM ?
Answer:
[tex](4 \times 1)+(3 \times \frac{1}{100})+(5 \times \frac{1}{1000})[/tex]
Step-by-step explanation:
Given : MARIA WALKED 4.035 KILOMETERS.
To Find :WHAT IS 4.035 WRITTEN IN AS EXPANDED FORM ?
Solution :
Number = 4.035
The numbers after decimals when read from first to last has positions tenth , hundredth , thousandth and so ...
The number before the decimal part has positions ones , tens , hundreds and so on when read from last to first
Now 4 is at ones place
0 is at tenth place
3 is at hundredth place
5 is at thousandth place
So, Expanded form : [tex](4 \times 1)+(3 \times \frac{1}{100})+(5 \times \frac{1}{1000})[/tex]
the sum of two numbers is 18 . the difference of the two numbers is -2 find the numbers
Factor out the coefficient of the variable. The expression 2.2x +4.4 factored is
You make a large bowl of salad to share with your friends. Your brother eats 1/3 of it before they come over. What fractional portion of the original bowl of salad does each friend receive?
Answer:
[tex]\frac{2}{3x}[/tex]
Where, x = total friends
Step-by-step explanation:
Given,
The part of the salad has eaten = [tex]\frac{1}{3}[/tex]
Total part of the salad = 1,
Thus, the remaining part of the salad = original part - part has eaten
[tex]=1-\frac{1}{3}[/tex]
[tex]=\frac{3-1}{3}[/tex]
[tex]=\frac{2}{3}[/tex]
If there are x friends,
Then the portion of the original bowl of salad received by each friend
[tex]=\frac{\text{Remaining part}}{\text{Total friends}}[/tex] [tex]=\frac{2}{3x}[/tex]
Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = ye−xi + e−xj + 2zk
To ascertain if a vector field is conservative or not, you need to calculate the curl of the field or integrate over the components of the vector field. If the curl is zero, it's conservative. If the curl isn't zero or an integral doesn't exist, the field is not conservative.
Explanation:To determine if the vector field f(x, y, z) = ye−xi + e−xj + 2zk is conservative, we need to find if there exists a function f such that f is the gradient (denoted by ∇) of f. This can be done by checking if the cross product of the vector field is equal to zero, which signifies that the field is conservative.
First, we calculate the curl (∇ x F) of the vector field, which gives us the derivatives of the field components. If the curl is zero, then the vector field is conservative. If it is not zero, this indicates that the vector field is non-conservative.
Alternatively, we can integrate over the components of the vector field to try and find a potential function. If an integral exists, then we can say that the vector field is conservative.
However, if it fails these conditions, then the vector field is not conservative and the function f does not exist for it (dne). Thus, in the case where the vector field is not conservative, enter 'dne'.
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A school conference room can seat a maximum of 83 people. The principal and two counselors need to meet with the school’s student athletes to discuss eligibility requirements. If each student must bring a parent with them, what is the maximum number of students that can attend each meeting?
Answer:
The maximum number of students that can attend each meeting is:
40
Step-by-step explanation:
A school conference room can seat a maximum of 83 people.
The principal and two counselors need to meet with the school’s student athletes to discuss eligibility requirements.
Let there be x students,there will be x parents
that means x+x+3≤83
Subtracting both sides by 3,we get
2x≤80
dividing both sides by 2,we get
x≤40
Hence, the maximum number of students that can attend each meeting is:
40
The Perez family has a rectangular fish tank how much water will the tank hold if it's length is 3 feet, its width is 2 feet and its height is 3 feet
A) NONE
B) 18 cubic feet
C) 3 cubic feet
D) 6 cubic feet
E) 9 cubic feet
Suppose e(x) = 3 and e[x(x − 1)] = 27.5. (a) what is e(x2)? [hint: first verify that e[x(x − 1)] = e[x2 − x] = e(x2) − e(x).]
determine the quadratic function of f whose graph is given. The vertex is (1,-3) and the y-intercept is -2
whats the nearest tenth to -61 square root
You've got 60 homework problems to do and it took you 10 minutes to do eight of them. At that rate, how long will it take?
solve 6[4x(72-63)divided3]
Find the missing number of each unit rate
The slope of a line is 1, and the y-intercept is -1. What is the equation of the line written in slope-intercept form?
y = x - 1
y = 1 - x
y = -x - 1
Marco comma roberto comma dominique comma and claricemarco, roberto, dominique, and clarice work for a publishing company. the company wants to send two employees to a statistics conference. to be fair, the company decides that the two individuals who get to attend will have their names drawn from a hat. this is like obtaining a simple random sample of size 2. (a) determine the sample space of the experiment. that is, list all possible simple random samples of size n equals 2n=2. (b) what is the probability that marco and robertomarco and roberto attend the conference? (c) what is the probability that dominiquedominique attends the conferenceattends the conference?
Answer:
Yes
Step-by-step explanation:
ye mom ye mom lolololol
What is the volume of a paperback book 21 cm tall, 12 cm wide, and 3.5 cm thick?
The volume of a cuboid is given as length × width × height thus the volume of the paperback book will be 882 cm².
What is volume?Volume is the scalar quantity of any object that specified occupied space in 3D.
Volume has units of cube example meter³,cm³, etc.
Given a paperback book shape as cuboid
Length(tall) = 21 cm
Width(wide) = 12cm
Height (thick) = 3.5 cm
The volume of the cuboid is given as
Volume = length × height × width.
Volume = 21 × 12 × 3.5
Volume = 882 cm²
Hence" The volume of a cuboid is given as length × width × height thus the volume of the paperback book will be 882 cm²".
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Allison drive 30 mph through the city and 55 mph on the New Jersey Turnpike she drove 90 miles from battery Park to the Jersey shore how much of the time was city driving if she needs about 1.2 hours on the turnpike
WILL GIVE BRAINEST
categorize the graph as liner increase...
6578 divided 34
20\5
Answer:
263.56 :)
Step-by-step explanation:
6-1. let x have a poisson distribution with a mean of 4. find (a) p(2≤x≤5). (b) p(x≥3). (c) p(x≤3).
Amanda can jog 18 miles in 5 hours. At this rate, how many miles can Amanda jog in 4 hours?
we know that
Amanda can jog [tex]18[/tex] miles in [tex]5[/tex] hours
so
by proportion
Find the number of miles that Amanda can jog in [tex]4[/tex] hours
[tex]\frac{18}{5} \frac{miles}{hours} =\frac{x}{4} \frac{miles}{hours} \\ \\5*x=18*4 \\ \\x=72/5 \\ \\ x=14.4\ miles[/tex]
therefore
the answer is
[tex]14.4\ miles[/tex]
Find the volume of a rectangular block of ice 3 feet by 6 1/3 and 1 1/2 feet
Final answer:
The volume of the rectangular block of ice is 85.5 cubic feet. This was found by converting the mixed numbers into improper fractions, then multiplying the length, width, and height together using the formula volume = length × width × height.
Explanation:
To find the volume of a rectangular block of ice with the given dimensions, we simply need to multiply the length, width, and height together. The formula to calculate volume is Volume = length × width × height. First, however, we need to convert the mixed numbers into improper fractions so we can multiply them easily.
The length is given as 6 1/3 feet, which can be converted to an improper fraction: (6 × 3) + 1 = 19/3 feet. The height is given as 1 1/2 feet, which is (1 × 2) + 1 = 3/2 feet.
Now, to find the volume, we multiply these dimensions with the width, which is 3 feet.
Volume = (19/3) feet × 3 feet × (3/2) feet
The feet × feet × feet will give us cubic feet.
Multiplying these together:
Volume = (19 × 3 × 3) / (3 × 2) cubic feet
Volume = 171/2 cubic feet or 85.5 cubic feet
Thus, the volume of the block of ice is 85.5 cubic feet.
The measure of arc XZ is 28° what is the measure of XYZ
a neighborhood garden that is 2/3 of an acre is to be divided 4 equal-size sections
a fan has 5 equally spaced blades. what is the least number of degrees that can rotate the fan onto self?
Determine whether the given function is linear. if the function is linear, express the function in the form f(x)
The given function [tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex] is indeed linear, and it can be expressed as f(x) = x in the standard linear form.
Let's break down the analysis of the given function[tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex].
1. Initial Expression:
[tex]\[ f(x) = \frac{5}{5} \cdot x \][/tex]
2. Simplify the Fraction:
[tex]\[ \frac{5}{5} \][/tex] simplifies to 1, so the expression becomes:
[tex]\[ f(x) = 1 \cdot x \][/tex]
3. Multiplication by 1:
Multiplying any value by 1 does not change the value, so the expression further simplifies to:
f(x) = x
4. Linear Form:
The function is now in the form f(x) = ax + b with a = 1 and b = 0:
[tex]\[ f(x) = 1 \cdot x + 0 \][/tex]
Therefore, the given function [tex]\(f(x) = \frac{5}{5} \cdot x\)[/tex] is indeed linear, and it can be expressed as f(x) = x in the standard linear form.
Complete Question: Determine whether the given function is linear. if the function is linear, express the function in the form f(x) = ax + b. (if the function is not linear, enter not linear.)
f(x) = 5 / 5 x