Let F(x) = f(x ^ 5) * ndG(x) = (f(x)) ^ 5 You also know that a ^ 4 = 3 , f(a) = 2 , f^ prime (a)=13 , f^ prime (a^ 5 )=2
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Given the functions:
[tex]\begin{gathered} F(x)=f(x^5) \\ \\ G(x)=(f(x))^5 \end{gathered}[/tex]Additionally, we know that:
[tex]\begin{gathered} a^4=3 \\ f(a)=2 \\ f^{\prime}(a)=13 \\ f^{\prime}(a^5)=2 \end{gathered}[/tex]We use the chain rule to find the derivatives of F(x) and G(x):
[tex]\begin{gathered} F^{\prime}(x)=f^{\prime}(x^5)\cdot(x^5)^{\prime}=5x^4\cdot f^{\prime}(x^5) \\ \\ G^{\prime}(x)=5(f(x))^4\cdot f^{\prime}(x) \end{gathered}[/tex]Finally, evaluating for x = a and using the results stated before:
[tex]\begin{gathered} F^{\prime}(a)=5\cdot a^4\cdot f^{\prime}(a^5)=5\cdot3\cdot2 \\ \\ \therefore F^{\prime}(a)=30 \end{gathered}[/tex][tex]\begin{gathered} G^{\prime}(a)=5\cdot(f(a))^4\cdot f^{\prime}(a)=5\cdot2^4\cdot13 \\ \\ \therefore G^{\prime}(a)=1040 \end{gathered}[/tex]Related Questions
5. John had 2 Snickers bars for every 4 Kit-Kit bars. If John had a total of 16 candy bars,how many Snickers bars did he have?as
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The length of a rectangular room is 5 yards more than the width. If the area is 300 yd2, find the length and the width of the room.
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Okay, here we have this:
Considering that the area of a rectangle is:
Area=length*width
Replacing we obtain:
300=(5+x)*x
300=5x+x²
0=5x+x²-300
0=(x-15)(x+20)
This mean that:
x-15=0 or x+20=0
x=15 or x=-20
And considering that the distances are positive we are left with the first solution, x=15; this mean that:
Width=15 yd
Length=(15+5) yd=20 yd.
Finally we obtain that the width is 15 yd and length is 20 yd.
Aviation A plane leaves an airport and flies south at 180 mph.
Later, a second plane leaves the same airport and flies south at
450 mph. If the second plane overtakes the first one in 12 hours,
how much of a head start did the first plane have?
Answers
The first plane had a head start of 432 minutes.
The speed, time and distance of any entity can be related by the following expression as Distance = Speed × Time. The speed of the first plane is 180 mph and the time given is 12 hours whereas the speed of second plane is 450 mph. Now, the distance travelled by plane A in time 12 hours is given by
Distance = Speed × Time
Distance = 180 × 12
Distance = 2160 miles.
The second also flies 2160 miles to overtake the first plane. It does this at a rate of 450 mph. So, the time taken for it to fly will be
Time = Distance/Speed
Time = 2160/450
Time = 4.8 hours
Since, 1 hour = 60 minutes, Therefore, 4.8 hours = 4.8×60 = 288 minutes. Now, since the first plane flies for 12 hours = 12×60 = 720 minutes. So, the head start = 720 minutes - 288 minutes = 432 minutes.
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HELP NOW PLS !!! 100 POINTS
The points (5,8) and (10,2) fall on a particular line. What is its equation in point-slope form?
Use one of the specified points in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.
Answers
Answer:
[tex]y-8=-\dfrac{6}{5}(x-5)[/tex]
Step-by-step explanation:
Define the given points:
(x₁, y₁) = (5, 8)(x₂, y₂) = (10, 2)First find the slope of the line by substituting the given points into the slope formula:
[tex]\implies \textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{2-8}{10-5}=-\dfrac{6}{5}[/tex]
To find the equation in point-slope form, simply substitute the found slope and one of the given points into the point-slope formula:
[tex]\implies y-y_1=m(x-x_1)[/tex]
[tex]\implies y-8=-\dfrac{6}{5}(x-5)[/tex]
4. Find the value of p if 2P=2^2 p-7
Answers
Answer:
P = 7
Step-by-step explanation:
[tex]{ \tt{ {2}^{p} = {2}^{2p - 7} }} \\ [/tex]
- From the law of indices; If an index has same base, then the powers are equal.
[tex]{ \boxed{ \rm{ \blue{ ({x}^{a} = {x}^{b}) \rightarrow{ \red{a = b}} }}}}[/tex]
[tex]{ \tt{p = 2p - 7}} \\ { \tt{p -2 p = - 7}} \\ { \tt{p = 7}}[/tex]
OR:
Applying logarithms can also be borrowed;
[tex]{ \tt{ log( {2}^{p} ) = log( {2}^{2p - 7} ) }} \\ \\ { \tt{p log(2) = (2p - 7) log(2) }} \\ \\ { \tt{ \frac{p log(2) }{ log(2) } = \frac{(2p - 7) log(2) }{ log(2) } }} \\ \\ { \tt{p = 2p - 7}} \\ \\ { \tt{p - 2p = - 7}} \\ \\ { \tt{p = 7}}[/tex]
3. Consider the following system of equations.Line 1: 2x - y = -3Line 2: -6x - 2y = -6Part A:Is (0,3) a solution to Line 1? Explain your answer.Part B:Is coordinate (0, -3) is a solution to Line 2? Explain your answer.Part C:What are the slopes of Linel and Line 2?Part D:What are the y-intercepts of Line 1 and Line 2?
Answers
Part A
replace (0,3) on (x,y)
[tex]\begin{gathered} 2(0)-(3)=-3 \\ 0-3=-3 \\ -3=-3 \end{gathered}[/tex]the equivalence is correct so (0,3) is a solution of the first equation
Part B.
replace (0,-3) on (x,y)
[tex]\begin{gathered} -6(0)-2(-3)=-6 \\ 0-(-6)=-6 \\ 6=-6 \end{gathered}[/tex]the equivalence is incorrect so (0,-3) isnt a solution of the second equation
Part C
To find the slope we need to solve each expresion and take the coefficient of x
first equation
[tex]\begin{gathered} 2x-y=-3 \\ y=2x+3 \end{gathered}[/tex]the slope is 2
second equation
[tex]\begin{gathered} -6x-2y=-6 \\ 2y=-6x+6 \\ y=-3x+3 \end{gathered}[/tex]the slope is -3
Part D
the y-intercept is the constant without variable on each equation
first equation
[tex]y=2x+3[/tex]the y-intercept is 3
second equation
[tex]y=-3x+3[/tex]the y-intercep is 3 too
The Graph
we need two points of the line and join by a right infinite line
first equation
the points (0,3) and (-3/2,0) belong to the line 1
second equation
the points (0,3) and (1,0) belong to the line 2
Solution of the system
we can note the two lines trought the point (3,0) so this is the solution and we can check matching the equations and solving x
[tex]\begin{gathered} 2x+3=-3x+3 \\ 2x+3x=3-3 \\ 5x=0 \\ x=0 \end{gathered}[/tex]and replace x=0 on any equation to solve y I will use the first equation
[tex]\begin{gathered} y=2x+3 \\ y=2(0)+3 \\ y=3 \end{gathered}[/tex]so the solution point is (0,3)
Jill works at a coffee shop on weekends. Every now and then, a customer will order a hot tea and ask Jill to surprise them with the flavor. The teas are categorized by flavor and caffeine level. Mint Fruity Caffeine-free 2 7 Caffeinated 5 5 What is the probability that a randomly selected tea is caffeinated or mint? Simplify any fractions.
Answers
The grand total is given by
[tex]n=2+7+5+5=19[/tex]so, the probability of Caffeinated is
[tex]P(Caffeinated)=\frac{5}{19}+\frac{5}{19}=\frac{10}{19}[/tex]the probability of mint is
[tex]P(\min t)=\frac{2}{19}+\frac{5}{19}=\frac{7}{19}[/tex]and the probability of the intersection is
[tex]P(Caffeinated\cap\min t)=\frac{5}{19}[/tex]Then, the probabilty of the union is given by
[tex]undefined[/tex]Solve the following system of equations using the substitution method.
–6x + 2y = 8
y = 3x + 4
Answers
Answer:
Infinite solutions or some courses say all real numbers
Step-by-step explanation:
-6x + 2(3x + 4) = 8 substitute 3x + 4 for y
-6x + 6x + 8 = 8 Distribute the 2
8=8 The x's cancel out leaving a true statement. This means that there are infinite solutions.
Beginning with the general form of a quadratic equation. ax^2+bx+c=0, solve for x by using the completing the square method, thus deriving the quadratic formula. To earn full credit be sure to show all steps/calculations. You may want to do the work by hand and upload a picture of that written work rather than try to type it out.
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to solve ax^2 + bx + c = 0 using completing the square method
divide all terms by a so as to reduce the coefficient of x^2 to 1
x^2 + bx/a + c/a = 0
subtract the constant term from both sides of the equation
x^2 + bx/a = -c/a
to have a square on the left sie the third term (constant) should be
(b/2a)^2
so add that amount to both side
x^2 + bx/a + (b/2a)^2 = (b/2a)^2 - c/a
rewrite the left side as a square
(x + (b/2a))^2 = (b/2a) - c/a
take the square root of both sides
x + (b/2a) = + square root of (b/2a)^2 - c/a
subtract the constant term on the left side from both sides
[tex]\begin{gathered} x\text{ = }\pm\sqrt[]{(\frac{b}{2a}})^2\text{ - c/a - (b/2a)} \\ x\text{ = -b }\pm\sqrt[]{\frac{b^2\text{ - 4ac}}{2a}} \end{gathered}[/tex]An expert witness for a paternity lawsuit testifies that the length of a pregnancy is normally distributed with a mean of
280 days and a standard deviation of 13 days. An alleged father was out of the country from 242 to 301 days before the birth
of the child, so the pregnancy would have been less than 242 days or more than 301 days long if he was the father. The birth
was uncomplicated, and the child needed no medical intervention. What is the probability that he was NOT the father?
Calculate the z-scores first, and then use those to calculate the probability. (Round your answer to four decimal places.)
What is the probability that he could be the father? (Round your answer to four decimal places.)
Answers
1. The z scores in the question are - 2.92 and 1.615
2. The probability that he is the father = 0.054905
How to solve for the probability and the z scoreThe z score for the 242 days
= 242 - 280 / 13
= -2.92
The z score for the 30 days
= 301 - 280 / 13
= 1.615
Next we have to solve for The probability that he is not the father
this is written as
p(242 < x < 301)
p value of -2.92 = 0.00175
p value of 1.615 = 0.946845
Then we would have 0.946845 - 0.00175
= 0.945095
The probability that he is the father is given as 1 - probaility that he is not the father of the child
= 1 - .945095
= 0.054905
The probability that he is the father is 0.054905
What is probability?This is the term that is used in Statistics and also in the field of mathematics to explain the chances and the likelihood of an event occurring.
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ASAP I NEED HELP WITH THIS PROBLEM AND WILL GET THE BRAINLIEST FOR THE CORRECT ANSWER
Answers
Answer: (x, y) -> (x, -y)
Step-by-step explanation:
1) You can easily find the transformation by substituting one point on the figure.
For this example, I will substitute S and S' points. (4, 1) and (4, -1)
2) Replace the numbers with x and y.
Set the numbers equal. They are already equal so no change.
(4, 1) -> (4, -1)
Replace with X and Y
(x, y) -> (x, -y)
Please help and round to the nearest minute if needed
Answers
Solution
For this case we have the following angle:
30 1/6 º
and then we need to convert to degrees and minutes so we can do this:
1 º= 60 min
then 1/6º* (60min/ 1º)= 10 min
Then the answer is:
30º 60'
John purchased 4
apples for $1.25
each and 1 orange
for 2.49 How
much does he
spend in all?
Answers
5+2.49= 7.49
Answer John spent 7.49$
Answer:
$7.49
Step-by-step explanation:
$1.25 x 4 = $5
1 x $2.49 = $2.49
5 + 2.49 = $7.49
Use the graph to complete the statement. O is the origin. R(y-axis) o R(y=x): (2,3)A. (-2, -3)B. (-3, 2)C. (3, -2)D. (2, -3)
Answers
Answer:
C. (3, -2)
Explanation:
First, we need to reflect the point (2, 3) across the line y = x. To reflect this point, we need to find another point that is at the same distance but on the opposite side of the line, so the reflection is
Therefore, the reflection is the point (3, 2)
Then, reflect (3, 2) across the y-axis, to get:
So, the answer is
C. (3, -2)
Choose if each statement is True or False.
{(2, 2), (3, 2), (4, 2), (6, 2)} is a function:
{(-1, 5), (0, 8), (3, 12), (6, 21)} is a function:
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Answer:
The first one in red is a function. The second one in blue is not a function.
Step-by-step explanation:
Using the vertical line test, if you were to draw a vertical line and move the line from left to right, it should not have two points of intersection (if the vertical line intersects the relation more than one, then the relation is not a function).
What is the GCF of 12 and 24
Answers
Answer:
12
Step-by-step explanation:
The GCF of 12 and 24 is 12. To calculate the most significant common factor of 12 and 24, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; characteristics of 24 = 1, 2, 3, 4, 6, 8, 12, 24) and choose the most significant factor that exactly divides both 12 and 24, i.e., 12.
12(1,2,3,4,6,12)
24(1,2,3,4,6,8,12,24)
Answer choicesReflection:1. reflect in the x-axis2. No reflectionStretch/Compress:1. No stretch nor compression2. Vertical Stretch of 2Horizontal Translation:1. Shift 6 units left2. Shift 5 units left3. Shift 6 units right4. Shift 5 units rightVertical Translation:1. Shift 5 units up2. Shift 6 units down3. Shift 6 units up4. Shift 5 units down
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First, the parent function is translated 5 units to the left, then it is reflected over the x-axis, and finally, it is translated 6 units down.
Answer:
Reflection: reflect in the x-axis.
Stretch: No stretch nor compression.
Horizontal Translation: Shift 5 units left.
Vertical Translation: Shift 6 units down.
(12-1) (-2-3) slope p l z
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[tex]m = \frac{y2 - y1}{x2 - x1} \\ m = \frac{ - 3 - ( - 1)}{ - 2 - 12} \\ m = \frac{ - 3 + 1}{ - 14} \\ m = \frac{ - 2}{ - 14} \\ m = \frac{1}{7} [/tex]
ATTACHED IS THE SOLUTION..I also provided you with the formula used to get the gradient.
PLEASE HURRY
What is the quotient of (−152) ÷ (−19) ÷ (−4)?
Answers
Answer:
-2
I did the math and it came out -2
Use point-slope form to write the equation of a line that passes through the point (-5,7)(−5,7) with slope -5−5
Answers
The equation of the line that passes through the point (-5,7) with slope -5 in the point-slope form is 5x + y = -18 .
The Point - Slope form of the line passing through (x₁,y₁) with slope m is given by the equation
(y-y₁)= m(x-x₁)
In the question ,
it is given that the required line passes through the point(-5,7) and have the slope = -5 .
the point is (-5,7)
so x₁= -5 and y₁=7 and m = -5
Substituting the value in the equation of point slope form , we get
(y-y₁)= m(x-x₁)
(y-7)= (-5)(x-(-5))
simplifying further , we get
(y-7)= (-5)(x+5)
y-7 = -5x -25
5x + y = -25 +7
5x + y = -18
Therefore , the equation of the line that passes through the point (-5,7) with slope -5 in the point-slope form is 5x + y = -18 .
The given question is incomplete , the complete question is
Use point-slope form to write the equation of a line that passes through the point (-5,7) with slope -5 .
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Solve the inequality for x and identify the graph of its solution. 4[x+ 2] < 8
Answers
Answer:
x < 0
Step-by-step explanation:
4(x + 2) < 8
4x + 8 < 8
4x < 0
x < 0
◀━━━━━|──>
0
Let f(a) = x^2 + 5.a) Find the y-value when x = 0.The y-value, output value is ___b) Find the y-intercept, when x = 0.The y-intercept is ___c) Find the x-values, when y = 46.The x-values are ____
Answers
To solve a, we need to replace x = 0 in the formula of the function:
[tex]\begin{cases}f(x)=x^2+5 \\ x=0\end{cases}\Rightarrow f(0)=0^2+5=5[/tex]The y value when x = 0 is 5.
b is asking the same as a but in a different way. The y-intercept of a function is when x = 0, we just calculated that. The point of y-intercept is (0, 5)
Finally, to solve c, we need to find the values of x that gives us a value of f(x) = 46:
[tex]f(x)=46\Rightarrow46=x^2+5[/tex]Then solve:
[tex]\begin{gathered} x^2=46-5 \\ x=\pm\sqrt[]{41} \end{gathered}[/tex]Remember that we must that plus-minus the value when we take square root. ± √41 is the answer to c.
i’m trying to find the slope and y intercept of number 2.. can someone help me please. thank you (:
Answers
According to the problem,
• Jennifer is 20 miles North.
,• The rate is 55 miles per hour.
Remember that rate of change refers to the slope.
Therefore, the slope is 55.
On the other hand, the y-intercept is the initial condition of the problem since Jennifer started 20 miles North, then the y-intercept is (0,20).
Hello, I need help with this practice problem, thank you!
Answers
In order to find the distance between the given points, use the following formula:
[tex]d=\sqrt[]{(x_2-x_1_{}^{})^2+(y_2-y_1)^2}[/tex]where (x1,y1) and (x2,y2) are the coordinates of the points.
In this case, you have:
(x1,y1) = K(1,-1)
(x2,y2) = F(6,-9)
Replace the previous values of the parameters into the formula for d and simplify:
[tex]\begin{gathered} d=\sqrt[]{(6-1)^2+(-9-(-1))^2} \\ d=\sqrt[]{(5)^2+(-9+1)^2} \\ d=\sqrt[]{25+(-8)^2}=\sqrt[]{25+64} \\ d=\sqrt[]{89} \end{gathered}[/tex]Hence, the distance between K and F points is √89.
Janet is getting balloons for her grandmother's birthday party. She wants each balloon string to be 12 feet long. At the party store, string is sold by the yard. If Janet wants to get 84 balloons, how many yards of string will she need?
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Using conversion factors we can conclude that Janet needs 336 yards of string.
What do we mean by conversion factor?A conversion factor is a number that is used to multiply or divide one set of units into another. If a conversion is required, it must be done using the correct conversion factor to get an equal value. For instance, 12 inches equals one foot when converting between inches and feet.So, yards of string are needed:
1 balloon string will be 12 feet longSo, 84 balloons will have:12 × 84 = 1,008 feet stringsWe know that:
1 feet = 0.3333 yardsThen, 1008 feet = 336 yardsTherefore, using conversion factors we can conclude that Janet needs 336 yards of string.
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Which postulate or theorem proves that these two triangles are
congruent?
• SAS Congruence Postulate
• ASA Congruence Postulate
O AAS Congruence Theorem
R
O HL Congruence Theorem
Answers
According to the Angle-Side-Angle Postulate (ASA), if two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, the two triangles are congruent.
What is postulate?A postulate is a statement that is assumed to be true in the absence of proof. A theorem is an unprovable true statement. Six postulates and theorems that can be proven from them are listed below. A statement, also known as an axiom, that is assumed to be true in the absence of proof. Postulates are the fundamental building blocks from which lemmas and theorems are derived. Euclidean geometry, for example, is built around five postulates known as Euclid's postulates. A postulate is a statement accepted without evidence. A postulate is another name for an axiom.To learn more about postulate, refer to:
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given the following trig equations find the exact value of the remaining five trig functions.cos0 = 4/9 where sin0 < 0( sin, tan, csc, cot, sec)
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we have that:
[tex]\sin ^2\theta=1-\cos ^2\theta=1-\frac{16}{81}=\frac{65}{81}\rightarrow\sin \theta=-\frac{\sqrt[]{65}}{9}[/tex]having this we get that
[tex]\tan \theta=\frac{-\sqrt[]{65}}{4},\cot \theta=-\frac{4}{\sqrt[]{65}},\sec \theta=\frac{9}{4},\csc \theta=-\frac{9}{\sqrt[]{65}}[/tex]what is the diameter of a circle if the circumference is 18 cm
Answers
Kuta Software - Infinite Algebra 2 Graphing Absolute Value Equations Graph each equation. 1-1-11 Name Date > 1512020 2) y-lx-a 13
Answers
we have the equation
[tex]y=-3\lvert-2x+4\rvert+3[/tex]using a grahing tool
see the attached figure
Jessica has a barrel to fill with water. The barrel is 24 inches high with a radius of 12 inches. She is using a cup to fill the the barrel. The cup has a height of 6 inches and diameter of 4 inches. How many full cups will she need in order to fill the barrel?
Answers
SOLUTION.
The barrel and the cub are both cylinders. To find how many cups that will fill the barrel, we find the volumes of both the cup and the barrel and divide that of the barrel by the cup
Volume of a cylinder is given as
[tex]\begin{gathered} \text{Volume = }\pi r^2h,\text{ r is radius and h is height of the cylinder } \\ radius\text{ of the barrel = }12,\text{ height = 24} \\ \text{Volume of barrel = }\pi r^2h \\ \text{Volume of barrel = 3.14}\times12^2\times24 \\ \text{Volume of barrel = }10851.84inch^3 \end{gathered}[/tex]Volume of the cub becomes
[tex]\begin{gathered} \text{radius of cup = }\frac{4}{2}=2 \\ \text{Volume of barrel = }\pi r^2h \\ \text{Volume of cup = }3.14\times2^2\times6 \\ \text{Volume of cup = }75.36inches^3 \end{gathered}[/tex]Number of cups become
[tex]\frac{10851.84}{75.36}\text{ = 144 cups }[/tex]