Answer:
[tex]\displaystyle f'(1) = \frac{3}{2}[/tex]
[tex]\displaystyle f''(1) = \frac{3}{4}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightAlgebra II
Exponential Rule [Rewrite]: [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]Exponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Calculus
Derivatives
Derivative Notation
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Step-by-step explanation:
Step 1: Define
f(x) = x√x
f'(1) is x = 1 for 1st derivative
f''(1) is x = 1 for 2nd derivative
Step 2: Differentiate
[1st Derivative] Product Rule: [tex]\displaystyle f'(x) = \frac{d}{dx}[x]\sqrt{x} + x\frac{d}{dx}[\sqrt{x}][/tex][1st Derivative] Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle f'(x) = \frac{d}{dx}[x]\sqrt{x} + x\frac{d}{dx}[x^{\frac{1}{2}}][/tex][1st Derivative] Basic Power Rule: [tex]\displaystyle f'(x) = (1 \cdot x^{1 - 1})\sqrt{x} + x(\frac{1}{2}x^{\frac{1}{2}-1})[/tex][1st Derivative] Simply Exponents: [tex]\displaystyle f'(x) = (1 \cdot x^0)\sqrt{x} + x(\frac{1}{2}x^{\frac{-1}{2}})[/tex][1st Derivative] Simplify: [tex]\displaystyle f'(x) = \sqrt{x} + x(\frac{1}{2}x^{\frac{-1}{2}})[/tex][1st Derivative] Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle f'(x) = \sqrt{x} + x(\frac{1}{2x^{\frac{1}{2}}})[/tex][1st Derivative] Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle f'(x) = \sqrt{x} + x(\frac{1}{2\sqrt{x}})[/tex][1st Derivative] Multiply: [tex]\displaystyle f'(x) = \sqrt{x} + \frac{x}{2\sqrt{x}}[/tex][2nd Derivative] Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle f'(x) = x^{\frac{1}{2}} + \frac{x}{2x^{\frac{1}{2}}}[/tex][2nd Derivative] Basic Power Rule/Quotient Rule [Derivative Property]: [tex]\displaystyle f''(x) = \frac{1}{2}x^{\frac{1}{2} - 1} + \frac{\frac{d}{dx}[x](2x^{\frac{1}{2}}) - x\frac{d}{dx}[2x^{\frac{1}{2}}]}{(2x^{\frac{1}{2}})^2}[/tex][2nd Derivative] Simplify/Evaluate Exponents: [tex]\displaystyle f''(x) = \frac{1}{2}x^{\frac{-1}{2}} + \frac{\frac{d}{dx}[x](2x^{\frac{1}{2}}) - x\frac{d}{dx}[2x^{\frac{1}{2}}]}{4x}[/tex][2nd Derivative] Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{\frac{d}{dx}[x](2x^{\frac{1}{2}}) - x\frac{d}{dx}[2x^{\frac{1}{2}}]}{4x}[/tex][2nd Derivative] Basic Power Rule: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{(1 \cdot x^{1 - 1})(2x^{\frac{1}{2}}) - x(\frac{1}{2} \cdot 2x^{\frac{1}{2} - 1})}{4x}[/tex][2nd Derivative] Simply Exponents: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{(1 \cdot x^0)(2x^{\frac{1}{2}}) - x(\frac{1}{2} \cdot 2x^{\frac{-1}{2}})}{4x}[/tex][2nd Derivative] Simplify: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{2x^{\frac{1}{2}} - x(\frac{1}{2} \cdot 2x^{\frac{-1}{2}})}{4x}[/tex][2nd Derivative] Multiply: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{2x^{\frac{1}{2}} - x(x^{\frac{-1}{2}})}{4x}[/tex][2nd Derivative] Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{2x^{\frac{1}{2}} - x(\frac{1}{x^{\frac{1}{2}}})}{4x}[/tex][2nd Derivative] Multiply: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{2x^{\frac{1}{2}} - \frac{x}{x^{\frac{1}{2}}}}{4x}[/tex][2nd Derivative] Simplify: [tex]\displaystyle f''(x) = \frac{1}{2x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{4x}[/tex][2nd Derivative] Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle f''(x) = \frac{1}{2\sqrt{x}} + \frac{\sqrt{x}}{4x}[/tex]Step 3: Evaluate
[1st Derivative] Substitute in x: [tex]\displaystyle f'(1) = \sqrt{1} + \frac{1}{2\sqrt{1}}[/tex][1st Derivative] Evaluate Roots: [tex]\displaystyle f'(1) = 1 + \frac{1}{2(1)}[/tex][1st Derivative] Multiply: [tex]\displaystyle f'(1) = 1 + \frac{1}{2}[/tex][1st Derivative] Add: [tex]\displaystyle f'(1) = \frac{3}{2}[/tex][2nd Derivative] Substitute in x: [tex]\displaystyle f''(1) = \frac{1}{2\sqrt{1}} + \frac{\sqrt{1}}{4(1)}[/tex][2nd Derivative] Evaluate Roots: [tex]\displaystyle f''(1) = \frac{1}{2(1)} + \frac{1}{4(1)}[/tex][2nd Derivative] Multiply: [tex]\displaystyle f''(1) = \frac{1}{2} + \frac{1}{4}[/tex][2nd Derivative] Add: [tex]\displaystyle f''(1) = \frac{3}{4}[/tex]Every week, Marla’s mom buys 15 liters of water at the supermarket. How many milliliters of water is this?
15,000
1,500
150
1000
Answer:
it is 15,000
Step-by-step explanation:
15,000 because there is 1,000 milliliters in one liters and 15 liters = 15,000 milliliters
Factor
2x3 + 6x2 - 36x
Answer:
23+62−36
2(3+32−18)
Step-by-step explanation:
What's the value of x
(3x-5) and (x+10)
5=10+n/6
5(6)=10+n(6)
30=10+n
-10 -10
Answer: sooooo....... the answer is n=20
Step-by-step explanation: did you really needed the answer for this?
Answer:
n = -30
Step-by-step explanation:
5 = 10 + n/6
-5 = n/6
n = -30
Joe has 16 pounds of fertilizer for his three gardens. He uses ⅜ of the fertilizer in his first garden and ¼ of the fertilizer in his second garden. What fraction of the fertilizer is left for the third garden? *
Answer:
3/8
Step-by-step explanation:
3/8 of 16 is 6
1/4 of 16 is 4
6+4 = 10
16-10 = 6
3/8 of 16 = 6
Find the missing angle according to the Triangle Sum Theorem.
Answer:
k = 140 degrees.
The missing INTERIOR angle is 40 degrees.
Step-by-step explanation:
The missing interior angle would be 40.
Triangle angle sum theorem states the 3 interior angles of a triangle sum to 180.
64+76+40 = 180
The exterior angle theorem states the exterior angle of a triangle is equal to the sum of the two remote interior angles of the triangle.
Exterior angle k = 64+76
k= 140
Edgenutiy which triangle is a 30 60 90
Answer:
yeah its a right triangle
Step-by-step explanation:
The high temperatures for a week are shown. 15°, 34°, 17°, 15°, 16°, 8° Which measure of center, mean or median, would BEST represent the data?
Hey there!
The mean would probably best represent the data.
This is because the mean of data is the average, which would show what the average temperature throughout the week was.
Have a super awesome day! :)
Someone help me please.
Answer:
I think the answer is 1,370 , not sure .
All of the items in a bicycle shop are on sale for 30% off the regular price. Rene spends $74.90 to buy a bicycle and a
helmet (without taxes). If the regular price of the bicycle is $85, which equation can be used to find the regular price of the
helmet, h?
0.3(74.90+ h) - 85
0.3(85+) - 74.90
0.7(74.90+h) - 85
0.7(85+1) - 74.90
Answer:
0.7(85+)-74.90
Step-by-step explanation:
Answer:
D - 0.7(85+)-74.90
Step-by-step explanation:
Got 100% on the quiz! Have a good day!! :)
An elliptical table is being cut from a rectangular piece of wood that is 12 feet by 10 feet. Legs of the table will be
attached at the foci. Which equation models the largest ellipse that can be cut from the rectangle and describes
where the legs will be attached?
Answer:
B on edge
Step-by-step explanation:
just got it wrong
Compare the areas. What is the ratio of the areas? SHOW YOUR WORK
Answer:
Ratio of the area of MIKE to TEAM = 9 : 1
Step-by-step explanation:
Ratio of similar shapes = square of the ratio of their corresponding sides
Since TEAM ~ MIKE, therefore the ratio of their areas = square of the height of MIKE : square of the height of TEAM
Ratio = 33² : 11²
Ratio = 1089 : 121
Ratio = 9 : 1
what are the special purpose books?
Answer:
hi Plz Subscribe Sameer Duos Channel
in OPQ, the measure of Q=90°, the measure of O=48°, and QO = 4.9 feet. Find
the length of PQ to the nearest tenth of a foot.
please help ne
∠ O ≈ 42.7°
Step-by-step explanation:
Using the cosine ratio in the right triangle
cos O = = = , thus
∠ O = ( ) ≈ 42.7° ( to the nearest tenth )
Can I have brainliest
Which one is greater 9/11 or 4/5
This is really easy please help
Answer:
9/11 = 45/55
4/5 = 44/55
Obviously, 45 is more than 44, so 9/11 is greater than 4/5.
Let me know if this helps!
Multiple three hundred and seventy by forty three
Step-by-step explanation:
15910
hope it helps
thanks a lot
Help please i'll give you brainliest
Prepare a graph with the X axis range from -300 to +100 °C and Y axis from 0 to 9 cm.
Answer: (Attached)
Graphic below is the answer to your question.
The range of F(x) = 6x3^x is all positive real numbers,
what is the volume of a hemisphere with a diameter of 41.6m, rounded to the nearest 10th of a cubic meter
Explanation is in the file
tinyurl.com/wtjfavyw
Tell whether c = 3 is a solution of c + 7 = 10.
a) Yes
b) No
Answer:
a) Yes
Step-by-step explanation:
Because, 3+7=10
Answer: yes
Explanation: In this problem, to determine whether c = 3
is a solution to the equation, we substitute 3 in for c.
So we have (3) + 7 = 10.
Simplifying the left side of the equation, (3) + 7 is 10.
So we have 10 = 10 which is a true statement so our answer is yes.
help me i am gonna give points to the people give me correct answer
Answer:
lol
Step-by-step explanation:
Answer:
Arjan should look for other cards that add up to $$94%
Step-by-step explanation:
I don't know it but I guess this one, let me know if it help
20% of the middle school students decided not to go skiing. How many middle school students are there if 40 stayed in the lodge?
Write a proportion for the situation.
Solve the problem. Show your work.
Answer:
Step-by-step explanation:
s(20/100)=40
20s=4000
s=200
So there are a total of 200 students.
Answer:
There are 200 middle school students
Step-by-step explanation:
The equation for this question can be: 40 is 20% of what. This can be simplified to 40 = 0.2x and then to get x by itself you divide 40 by 0.2 which is 200.
What is the median of this set of data?
362, 187, 211, 298, 331, 250, 347, 199
Answer:
274
Step-by-step explanation:
Hey There!
So if you didnt know the median is the middle number ( in value) of the set
So first we would like to arrange the numbers in order from least to greatest
187 , 199 , 211 , 250 , 298 , 331 , 347 , 362
Then we take the middle number(s)
if there is just one then that is the answer but if there is two than we add them together and divide that by 2
In this case there are two which would be 250 and 298
So we add them together
250+298=548
Then we just divide that by 2
[tex]\frac{548}{2} =274[/tex]
Therefore the median is 274
hope this helps :)
Step-by-step explanation:
l hope this answer is helpful to you
What is 0.64 divided by 248?
Answer:
0.00258064516
Step-by-step explanation:
Answer:
0.64 divided my 248 = 0.00258064516129
the problem is set up incorrectly what you should have put is 248 divided my 0.64 the answer to that is = 387.5
Your welcome!
Step-by-step explanation:
[tex]10 \times + 3y = 5 \\ \times - y = - 6[/tex]
10x+3y=5
x-y+3y=6
HELP ASAP I WILL GIVE BRAINLYIST
Answer:
1,728cm squared
Step-by-step explanation:
1. Break the Composite Figure into smaller shapes such as, rectangles, triangles, and squares.
2. Use the area formula for rectangles, triangles, and squares to find the area of each smaller shape.
3. Add the area of the smaller shapes together to get the area of the entire Composite Figure.
Hope this is correct and helps :[]).
Researchers conducted a study to determine whether the majority of community college students plan to vote in the next presidential election. They surveyed 650 randomly selected community college students and found that 55% of them plan to vote.
Which of the following are the appropriate null and alternative hypotheses for this research question?
A)
0: =0.50
: ≠0.50
B)
0: =0.50
: >0.50
C)
0: =0.55
: ≠0.55
D)
0: =0.55
: >0.55
Answer is:
B
Answer:
[tex]H_{0} = 0.5[/tex]
[tex]H_{a} > 0.5[/tex]
Option b
Step-by-step explanation:
Researchers conducted a study to determine whether the majority of community college students plan to vote in the next presidential election.
The null hypothesis is always an equalityu related to the expected, or tested value. In this question, the tested value is half of the students, a proportion of 0.5, that is: [tex]H_{0} = 0.5[/tex]
The alternate hypothesis is related to the test we are taking. In this question, we are testing if the majority of the students plan to vote, that is, if the proportion of the students that plan to vote is higher than 0.5. So [tex]H_{a} > 0.5[/tex]
So the correct answer is given by option B.
If YV = 28 in, find the length of the major arc VYX. Round to the nearest tenth ( one decimal place).
Answer:
[tex]VYX=84.3in[/tex]
Step-by-step explanation:
Given
[tex]YV = 28in[/tex]
[tex]\angle WZX = 15^\circ[/tex]
See attachment
Required
Determine the length of [tex]VYX[/tex]
First, calculate the major angle VZX
[tex]\theta = \angle VZX = 360 - \angle WZX[/tex]
[tex]\theta = \angle VZX = 360 - 15[/tex]
[tex]\theta = \angle VZX = 345[/tex]
[tex]\theta = 345[/tex]
The length of arc VYX, is:
[tex]L=\frac{\theta}{360} * 2\pi r[/tex]
Where
[tex]r = \frac{1}{2} * Diameter[/tex]
[tex]r = \frac{1}{2} * YV[/tex]
[tex]r = \frac{1}{2} * 28[/tex]
[tex]r = 14[/tex]
So:
[tex]L=\frac{\theta}{360} * 2\pi r[/tex]
[tex]L=\frac{345}{360} * 2 * \frac{22}{7} *14[/tex]
[tex]L=\frac{345}{360} * 2 * 22 *2[/tex]
[tex]L=\frac{345* 2 * 22 *2}{360}[/tex]
[tex]L=\frac{30360}{360}[/tex]
[tex]L=84.3in[/tex]
Hence, the length of VYX is:
[tex]VYX=84.3in[/tex]
12. Si R(x)=-5x2+x-2 y S(x)=-x - 3x, se
¿cuál es el resultado de R(x) + S(x) ?
Answer:
El resultado eres: [tex]R(x) + S(x) = -6x^2 - 2x - 2[/tex]
Step-by-step explanation:
Se dan las siguientes funciones
[tex]R(x) = -5x^2 + x - 2[/tex]
[tex]S(x) = -x^2 - 3x[/tex]
Cuál es el resultado de R(x) + S(x) ?
Se soman los fatores en comun. Entonces:
[tex]R(x) + S(x) = -5x^2 + x - 2 + (-x^2 - 3x) = -5x^2 + x - 2 - x^2 - 3x = -5x^2 - x^2 + x - 3x - 2 = -6x^2 - 2x - 2[/tex]
El resultado eres: [tex]R(x) + S(x) = -6x^2 - 2x - 2[/tex]