F (x) = axe ^ bx
F ‘ (x) = d/dx (axe ^ bx)
= a (d/dx) (xe ^ bx)
= a [(d/dx * x) e ^ bx + (x * d/dx * e ^ bx)]
= a [e ^ bx + bxe ^ bx]
= a [1 + bx] e ^ bx
For critical points, set f ‘ (x) = 0 and solve for x
a [1 + bx] e ^ bx = 0
= 1 + bx = 0
= bx = -1
X = -1/b
Given that f (x) has local max at 1/5, critical point x = 1/5
-1/5 = -1/b
Therefore b = -5
F(x) = axe ^ -5x
F(1/5) = a/5 e ^-1
1 = a/5e
Therefore, a = 5e
Answer:
The value of a and b is [tex]5e[/tex] and [tex]-5[/tex], respectively.
Step-by-step explanation:
The given function is [tex]f(x)=axe^{bx}[/tex].
The value of function at [tex]x=\dfrac{1}{5}[/tex] is [tex]f(\dfrac{1}{5})=1[/tex] and the function has a local maxima at [tex]x=\dfrac{1}{5}[/tex].
So, the first derivative of the function at [tex]x=\dfrac{1}{5}[/tex] will be zero.
Now, calculating for a and b.
[tex]f(x)=axe^{bx}\\f(\dfrac{1}{5})=a\left (\dfrac{1}{5}\right )e^{b\dfrac{1}{5}}\\1=\left (\dfrac{a}{5}\right )e^{\dfrac{b}{5}}[/tex]
Differentiate the function,
[tex]f(x)=axe^{bx}\\f'(x)=abxe^{bx}+ae^{bx}\\f'(1/5)=\dfrac{ab}{5}e^{\dfrac{b}{5}}+ae^{\dfrac{b}{5}}=0\\b=-5[/tex]
Solving for a as,
[tex]1=\left (\dfrac{a}{5}\right )e^{\dfrac{b}{5}}\\a=5e[/tex]
Therefore, the value of a and b is [tex]5e[/tex] and [tex]-5[/tex], respectively.
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Alinethatincludesthepoint(6, 3)hasaslopeof 2 . what isitsequationin slope -intercept form?
y=2x-3 is the equation of line that includes the point (6, 3) has a slope of 2
What is slope intercept form of line?The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
We need to find the equation of line that includes the point (6, 3) has a slope of 2.
y minus y one equal to m time of x minus x one.
(y-y₁)=m(x-x₁)
x₁=6,y₁=3 and m is 2
y-3=2(x-3)
Apply distributive property on right hand side.
y-3=2x-6
Shift -3 to the RHS side
y=2x-6+3
y=2x-3
Hence y=2x-3 is the equation of line that includes the point (6, 3) has a slope of 2
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In a paper airplane contest Amanda plane flew 19 ft farther than Darren plane Darren plane flew twice as far ad Rachel plane Rachel plane 20 ft .How far did Amanda plane fly
Answer:
59 ft
Step-by-step explanation:
Given;
Amanda plane flew 19 ft farther than Darren plane Darren plane flew twice as far ad Rachel planeRachel plane 20 ftThen Darren's plane must have flown (2 × 20ft) = 40 ft
Amanda's plane would have flown (40ft + 19ft) = 59 ft
Amanda's plane flew 59 ft far.
The distance of the paper plane thrown by Amanda is given by the equation A = 59 feet
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the distance of the paper plane thrown by Amanda be A
Now , the equation will be
The distance of the paper plane thrown by Rachel R = 20 feet
The distance of the paper plane thrown by Darren D = 2 x distance of the paper plane thrown by Rachel R
Substituting the values in the equation , we get
The distance of the paper plane thrown by Darren D = 40 feet
And , Amanda plane flew 19 ft farther than Darren plane Darren plane
Substituting the values in the equation , we get
The distance of the paper plane thrown by Amanda A = 19 + distance of the paper plane thrown by Darren D
On simplifying the equation , we get
The distance of the paper plane thrown by Amanda A = 19 + 40
The distance of the paper plane thrown by Amanda A = 59 feet
Hence , the distance is 59 feet
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The universal set is represented in a Venn diagram by...
A: Rectangle
B: Circle
C: Oval
How do I find 680% as a fraction mixed number or whole number in simplest form
To convert 680% to a fraction, write it as 680/100 and simplify to 34/5. This improper fraction can also be written as the mixed number 6 [tex]\frac{4}{5}[/tex].
To convert 680% to a fraction or mixed number, follow these steps:
First, write the percent as a fraction with a denominator of 100. So, 680% becomes 680/100.
Next, simplify the fraction by finding the greatest common divisor (GCD) of 680 and 100. The GCD of 680 and 100 is 20.
Divide both the numerator and the denominator by their GCD:
680 ÷ 20 = 34
100 ÷ 20 = 5
So, 680/100 simplifies to 34/5.
Finally, convert the improper fraction to a mixed number. To do this, divide the numerator by the denominator:
34 ÷ 5 = 6 (with a remainder of 4).
This gives us the mixed number 6 [tex]\frac{4}{5}[/tex].
Therefore, 680% as a fraction is 34/5 and as a mixed number is 6 [tex]\frac{4}{5}[/tex].
David is taking an economics course in which there will be 4 tests, each worth 100 points. he has scores of 89, 92, and 95 on the first three tests. he must make a total of at least 360 in order to get an
a. what scores on the last test will give david an a?
To get an A, David must score at least 85 on the last test.
David has taken three tests so far, with scores of 89, 92, and 95. To find out what score he needs on the last test to achieve a total of at least 360 points, we first calculate the total points he has accumulated from the first three tests:
[tex]\[ 89 + 92 + 95 = 276 \][/tex]
Now, we subtract this total from the minimum total points required for an A:
Therefore, David needs to score at least 84 points on the last test to reach a total of 360 points. However, since test scores are whole numbers, we round up to the nearest whole number that David can actually achieve on the test. Thus, David needs to score at least 85 on the last test to ensure he gets an A.
Bret saw a coat on sale for $28.55 it was originally priced at $33.59 what was the percent decrease in the price of the coat
Simplify -2[9 - (x + 7)].
Graph the function by the table. Is the function linear or nonlinear?
A. Nonlinear
B. Linear
Answer:
Option A - Non-linear
Step-by-step explanation:
Given : Graph the function by the table.
x y
1 12.3
2 19.6
3 26.6
4 34.2
5 41.5
To find : Is the function linear or nonlinear?
Solution :
The function is linear when there slopes are equal.
So, we find the slope of the points.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Taking point, [tex](x_1,y_1)=(1,12.3)[/tex] and [tex](x_2,y_2)=(2,19.6)[/tex]
[tex]m=\frac{19.6-12.3}{2-1}[/tex]
[tex]m=\frac{7.3}{1}[/tex]
[tex]m=7.3[/tex]
Taking point, [tex](x_1,y_1)=(2,19.6)[/tex] and [tex](x_2,y_2)=(3,26.6)[/tex]
[tex]m=\frac{26.6-19.6}{3-2}[/tex]
[tex]m=\frac{7}{1}[/tex]
[tex]m=7[/tex]
We have seen that slope between points are not equal.
Therefore, The given points are non-linear.
So, Option A is correct.
Name four coplanar points
The most efficient first step in the process to factor the trinomial 4x3-20x2+24x
A. factor out –1
B. factor out 4
C. factor out 4x
D. factor out (x – 3)
Factor out 4x from the given trinomial. Therefore, option C is the correct answer.
What are factors of polynomial?The factors are the polynomials which are multiplied to produce the original polynomial.
The given trinomial is 4x³-20x²+24x.
Here, 4x(x²-5x+6)
= 4x(x²-3x-2x+6)
= 4x[(x(x-3)-2(x-3)]
= 4x(x-3)(x-2)
Therefore, option C is the correct answer.
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On a coordinate plane, polygon GHIJ translates 8 units to the left to form polygon G 'H 'I 'J '. Which of the following equations is not necessarily true?
a) GH=G'H'
b) G'G=8 units
C) m
D) m
The equation which is not necessarily true is that G'H' = 8 units.
What is the equation?
There are many different ways to define an equation. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of 2 mathematical expressions.
A formula known as an equation uses the same sign to denote the equality of two expressions.
In another word, the equation must be constrained with some constraint.
Given that a polygon of GHIJ translates 8 units to the left to form polygon G 'H 'I 'J ' in the option it is necessary because GH is a side and G'H' will still be a side so they will be the same.
In option b G'G will be 8 units because the entire figure has shifted towards the left.
And the correct answer will be that G'H' = 8 units which are not true.
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Two mathematically similar frustums have heights of 20cm and 30cm.
The surface area of the smaller frustum is 450cm².
Calculate the surface area of the larger frustum.
The larger frustum has a surface area of 1516.875cm². This is calculated by first finding the ratio of the heights, then cubing this ratio and finally multiplying the result by the surface area of the smaller frustum.
Explanation:In Mathematics, similar frustums follow a rule in which the ratio of their dimensions is the cube root of their surface area ratio. Since the height of the larger frustum is 30cm, and the height of the smaller one is 20cm, their ratio is 30:20 or 1.5:1.
To find the ratio of the surface areas, you'll want to cube this ratio (1.5 ^ 3 = 3.375).
So, the surface area of the larger frustum is the surface area of the smaller frustum (which is 450cm² ) multiplied by this ratio. In mathematical terms, 3.375 * 450 = 1516.875cm².
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The legs of a trapezoid are parallel. True False
Answer:
false
Step-by-step explanation:
Please explain
Use a matrix to find solution to the system of equations
{-8x - 8y = -16
6x - 9y = -108
(-6,8)
(6,8)
(8,-6)
(6,-8)
Answer:
Step-by-step explanation:
the right answer is a (-6,8) where x=-6 and y=8
We're going to use Cramer's rule to resolve this question.
First of all, we're going to give a number to the equations:
(1) -8x-8y=-16
(2) 6x-9y=-108
where constant 1=c1=-16 and the constant 2=c2=-108
now we need to find the next determinants:
D: system determinant,Dx and Dy.
now we're going to find "D."
for that, we're going to take the coefficients of x and y in (1) and (2)
x y
D= [tex]\f\left[\begin{array}{ccc}-8&-8\\6&-9\end{array}\right][/tex]
this is how we need to put the matrix, using the numbers next to x and y in the (1) and (2) equation.
to solve this matrix, we need to multiplicate in X, and subtract the results; so for D we have:
(-8*-9)-(6*-8)
72-(-48)=
72+48=120
now we're doing the same process to find Dx and Dy, but in each case, we're going to replace x or y with the constants (the constants are the numbers right to the "=")
to find Dx we're going to use the values of c1 and c2 instead of x
C1&2 y
Dx= [tex]\f\left[\begin{array}{ccc}-16&-8\\-108&-9\end{array}\right][/tex]
Now we do the same as with D
(-16*-9)-(-108*-8)=
144-(864)
144-864=-720
and now for Dy=
x C1&2
Dy= [tex]\f\left[\begin{array}{ccc}-8&-16\\6&-108\end{array}\right][/tex]
= (-8*-108)-(6*-16)=
864-(-96)=
864+96=960
now the last step is to find x and y .
x= [tex]\frac{Dx}{D}[/tex] Y= [tex]\frac{Dy}{D}[/tex]
x= -720/120=-6
y=960/120=8
x=-6 y= 8
answer = (-6,8)
What is the slope ? Thanks
find the missing number 42/90 /15
Penny had $117, which is 9 times as much money as Kari had. How much money did Kari have?
Final answer:
Kari had $13. We found this by dividing Penny's amount, $117, by 9, since Penny's amount is 9 times greater than Kari's.
Explanation:
The question is asking us to determine how much money Kari had if Penny had $117, which is 9 times as much as what Kari had. To do this, we need to perform a division, because if Penny's amount is 9 times Kari's, we can find Kari's amount by dividing Penny's $117 by 9.
To solve for K, divide both sides of the equation by 9: K = $117 / 9.
Therefore, Kari had $13.
Please help!
What is the slope of the line passing through the points (1, −5) and (4, 1)?
A)−4/5
B)54
C)2
D)−2
What is the correct way to name these congruent triangles?
Which equations represent the line that is parallel to 3x − 4y = 7 and passes through the point (−4, −2)? Check all that apply.
y = -3/4x + 1
3x − 4y = −4
4x − 3y = −3
y – 2 = –3/4(x – 4)
y + 2 = 3/4 (x + 4)
Answer:
Option B and E
Step-by-step explanation:
A line is given as 3x - 4y = 7
We have to find the equation of a line parralel to the given line which passes through point (-4, -2).
Let the equation is y = mx + c
Line parallel to 3x - 4y = 7 will have same slope
3x - 4y = 7
-4y = 7 - 3x
4y = 3x - 7
y = [tex]\frac{1}{4}[/tex] (3x - 7 ) ≈ [tex]\frac{3}{4}[/tex]x - [tex]\frac{7}{4}[/tex]
So slope of the line will be m = [tex]\frac{3}{4}[/tex]
Since the given line passes through (-4, -2)
So we put the values in y = mx + c to get the value of c.
-2 = [tex]\frac{3}{4}[/tex] (-4 ) + c
-2 = -30 +c
c = 3 - 2 = 1
Therefore, equation will be
y = [tex]\frac{3}{4}[/tex]x + 1 ----------(1)
We further solve equation (1)
4y = 3x + 4
3x - 4y = -4 ---Option B
By solving the equation (1) in other way
y + 2 = [tex]\frac{3}{4}[/tex]x + 1 + 2
[tex]\frac{3}{4}[/tex]x + 3
y + 2 = [tex]\frac{3}{4}[/tex] (x+4) --Option E
Option B and E are the answers.
When two lines are parallel, the acute angles they form with a transversal are ________ to the obtuse angle?
Which type of graph would best display the data?
A bar graph would be the best way to summarize this information for display the data.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
We have to given that;
Andrea surveyed her classmates asking what type of pets they have.
Hence To show this on graph bar graph would be the best way to summarize this information for display the data.
Because, Bar graphs show comparisons between distinct categories and display changes over time.
Thus, A bar graph would be the best way to summarize this information for display the data.
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What does an exponent in front of a square root mean?
URGENT: Find two functions f(x) and g(x) such that f[g(x)] =x but g[f(x)] does not equal x.
Factor completely 5c5 + 60c4 + 180c3.
The length of a rectangle is 3 less than twice the width if the perimeter is 30 find the length and width
If a recipe calls for 8 pounds of trimmed celery, which has a 75 percent yeild,how many pounds of untrimmed celeray are needed
Final answer:
To find out how many pounds of untrimmed celery is needed to get 8 pounds of trimmed celery with a 75% yield, we divide the required trimmed weight by the yield percentage as a decimal. The calculation is 8 divided by 0.75, which equals approximately 10.67 pounds of untrimmed celery.
Explanation:
To determine the amount of untrimmed celery needed when a recipe requires 8 pounds of trimmed celery with a 75% yield, we use the yield percentage to find out how much the weight reduces during trimming. Yield percentage tells us how much of the original, or untrimmed, product remains after trimming. In this case, a 75% yield means that 75% of the untrimmed celery's weight will be left after trimming.
To find the amount of untrimmed celery, we set up the equation using the yield percentage (Y) of the untrimmed weight (U) that will result in the desired trimmed weight (T). The equation is T = U * Y/100. We can rearrange the equation to solve for U using the given values. So U = T / (Y/100), which gives us U = 8 / (75/100), resulting in U = 8 / 0.75.
After completing the division, we find that U equals 10.67. This means we need approximately 10.67 pounds of untrimmed celery to get 8 pounds of trimmed celery. Always remember to round according to the context, in most recipes a little more celery wouldn't be a problem, but it depends on the precision needed for the specific recipe.
A circle has a radius of 63 what is the arc length intercepted by a central angle that measures 2pi/9 radians
Help!!!
What equation is graphed in this figure?
y − 4 = −1/3 (x + 2)
y − 3 = 1/3 (x + 1)
y + 2 = −3 (x − 1)
y − 5 = 3 (x − 1)
The equation of the line is y + 2 = −3 (x − 1). The correct option is C.
What is an equation of the line?An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.
The general form of the equation of the line:-
y = mx + c
m = slope
c = y-intercept
Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )
The points from which the line passes are ( 0,1) and (1,-2). The slope of the line will be calculated as,
Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )
Slope = ( -2 - 1 ) / ( 1 )
Slope = -3
The equation of the line will be written as,
y - y₁ = m ( x - x₁ )
y +2 = -3 ( x - 1 )
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100 Points!!!
What number can you add to π to get a rational number?
−π
2
1/π
0
The answer is -π
If you add π + (-π), you get 0 which is a rational number.
Adding 2 only results in changing the 3.14... to a 5.14... which is still an irrational number.
Adding 1/π only results in adding 1/3.14... to π which equals 3.45... which is still an irrational number.
Adding 0 doesn't change the original value at all so its still an irrational number.
Therefore, -π is the only logical answer.
Best of Luck!
Answer:
- pi
Step-by-step explanation:
Pi is an irrational number. We need to add an equal and opposite irrational number to get a rational number. pi +-pi = 0
That is rational
Adding anything else will still leave us with pi in our answer which is irrational
2+pi
1/pi + pi =
0+pi
All have pi in the answer