The graphs of the function f (given in blue) and g (given in red) are plotted above. Suppose that u(x)=f(x)g(x) and v(x)=f(x)/g(x). Find each of the following:
u'(1) =
v'(1) =
This is a problem of calculus where the product and quotient rules are used to find the derivatives of functions u and v at x=1. The values need to be taken from the provided graph.
Explanation:This is a calculus question related to the product rule and the quotient rule of derivatives. First, let's explore the definitions: the product rule states that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second. The quotient rule states that the derivative of the quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared. From the graph, we have to find out the values of f(1), g(1), f'(1), and g'(1).
Then, we can calculate u'(1) by using the product rule: u'(1) = f'(1)g(1) + f(1)g'(1). In the same way, we can calculate v'(1) by using the quotient rule: v'(1) = [g(1)f'(1) - f(1)g'(1)] / [g(1)]^2. The specific values need to be derived from the provided graph
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(u’(1) = 5)
[tex](v’(1) = \frac{-13}{4})[/tex]
We have two functions:
(u(x) = f(x)g(x))
[tex](v(x) = \frac{f(x)}{g(x)})[/tex]
We need to find the derivatives of both (u(x)) and (v(x)) at (x = 1).
1. Derivative of (u(x)):
Using the product rule, we have:
[ u’(x) = [tex]v \cdot \frac{du}{dx} + u \cdot \frac{dv}{dx} ][/tex]
Where:
(v = g(x))
[tex](\frac{du}{dx})[/tex] is the derivative of (f(x))
[tex](\frac{dv}{dx})[/tex] is the derivative of (g(x))
From the graph:
At (x = 1), we have (f(1) = 3) and an estimated slope of the tangent line for (f’(1)[tex]\approx -2).[/tex]
Similarly, at (x = 1), we have (g(1) = 2) and an estimated slope of the tangent line for [tex](g’(1) \approx 3).[/tex]
Now let’s calculate (u’(1)):
[ u’(1) = [tex]g(1) \cdot f’(1) + f(1) \cdot g’(1)[/tex] = (2)(-2) + (3)(3) = 5 ]
2. Derivative of (v(x)):
Using the quotient rule, we have:
[ v’(x) = [tex]\frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ][/tex]
Substitute the values we found earlier:
(v = g(x) = 2)
[tex](\frac{du}{dx}[/tex] = f’(x) = -2)
[tex](\frac{dv}{dx} = g’(x) = 3)[/tex]
Now let’s calculate (v’(1)):
[tex][ v’(1) = \frac{(2)(-2) - (3)(3)}{2^2} = \frac{-13}{4} ][/tex]
Therefore:
(u’(1) = 5)
[tex](v’(1) = \frac{-13}{4})[/tex]
Your current account balance is $215. You have $322 of expenses each month. Your income is $444 per month. How long will it take you to accumulate a balance of $1000 in your account.
It take 6.434 month or 6 month and 13 days you to accumulate a balance of $1000 in your account.
What is word problem in mathematics?Word problem is a mathematical problem totally described in language that is frequently used as an educational tool.
Your current account balance is $215.
Your per month income is $444.
Your per month expense is $322.
Hence, your per month savings is = $444 - $322 = $122.
Let, it take x month you to accumulate a balance of $1000 in your account.
Then mathematically,
215 + 122 x = 1000
Rearrange the equation can be written that:
122 x = 1000 -215
After the subtraction, the result will be:
122 x = 785
Rearrange the equation can be written that:
x = 785/122
After the division, the result will be:
x = 6.434 = 6 month and 13 days.
Hence, it take 6.434 month or 6 month and 13 days you to accumulate a balance of $1000 in your account.
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Which function has the widest graph?
A. y=2x^2 B. y=x^2 C. y=0.5x^2 D.y=-x^2
Do you have to graph each one to find the answer? ...?
Slope is - 3 And (0,-4) is on the line
Which graph represents a reflection of f(x) = 2(0.4)x across the y-axis?
Answer: The correct option is A.
Explanation:
The given function is,
[tex]f(x)=2(0.4)^x[/tex]
To find the graph of this function after the reflecting across y-axis, first we have to find the graph of the equation.
The value of the function is 2 when x=0, so, the graph of given equation intersect the y-axis at 2.
In the equation [tex](0.4)^x[/tex]. Since [tex]0<0.4<1[/tex], so the given function is decreasing function.
[tex]f(x)\rightarrow 0 \text{ as }\rightarrow \infty[/tex]
[tex]f(x)\rightarrow \infty \text{ as }\rightarrow -\infty[/tex]
The value of f(x) is always positive, so the graph of f(x) is always above the x-axis. Thus, the graph must be above the x-axis after reflection across y-axis.
So, the option (2) and (4) and incorrect.
When we reflect the graph across the y-axis then,
[tex]f(x)\rightarrow \infty \text{ as }\rightarrow \infty[/tex]
[tex]f(x)\rightarrow 0 \text{ as }\rightarrow -\infty[/tex]
It means when x approaches to large negative number the f(x) approaches to 0 and when x approaches to large positive number the f(x) approaches to infinite.
Therefore, the correct option is show in first graph.
Answer:
a
Step-by-step explanation:
you are driving at a constant velocity of 100 km per hour. the first landmark you see is your favorite restaurant which happens to be 10 km from your house. which sequence describes the distances you cover at intervals of one hour, starting when you pass the restaurant?
Any ordered pair that makes all equations in a system of equations true is a(n) ...?
El area del cuadrado mide 25 pies cuadrados cuál es la longitud de un lado del cuadrado?
. To change the shape of a function, you need to _____ or compress it.
A. compress
B. stretch
C. shift
D. flip
Answer:
B. Strech
Step-by-step explanation:
Stretching can change the shape of something such as a function.
Two less than three times the width of a rectangle is equal to the length. The area of the rectangle is 65 square ft. What is the length of the rectangle?
Answer:
Length of the rectangle = 13 ft
Step-by-step explanation:
Let l be the length and w be the width of this rectangle.
Two less than three times the width of a rectangle is equal to the length, that is
3w - 2 = l---------------------------eqn 1
The area of the rectangle is 65 square ft
l x w = 65
Substituting eqn 1
(3w - 2) x w = 65
3w² - 2w -65 = 0
Solving quadratic eqn
[tex]w=\frac{-(-2)\pm \sqrt{(-2)^2-4\times 3\times (-65)}}{2\times 3}=\frac{2\pm \sqrt{4+780}}{6}\\\\w=\frac{2\pm \sqrt{784}}{6}\\\\w=\frac{2\pm 28}{6}\\\\w=\frac{2+28}{6}\texttt{ or }w=\frac{2-28}{6}\\\\w=5\texttt{ or }w=-4[/tex]
w cannot be negative
Hence width, w = 5 ft
Substituting in eqn 1
3 x 5 - 2 = l
l = 13 ft
Length = 13 ft
Length of the rectangle = 13 ft
help please?
solve for Y
a(y+c)=b(y-c)
solve for W
A=lw+wh+lh
...?
Geometry question help
15 pts will award brainliest if explained
Answer if you understand it only
a cheese pizza cost $8.75 each additional cost topping costs $1.25. if you order one pizza and have $12.50. how many toppings can you have on the pizza.
how to estimate 184÷9 in multiplication
The ordered pairs (1, 6), (2, 36), (3, 216), (4, 1296), and (5, 7776) represent a function. What is a rule that represents this function?
Answer:
The rule is [tex]\\ f(x) = 6^{x}[/tex]
Step-by-step explanation:
The resulting values are powers of 6:
[tex]\\ 6^{1} = 1[/tex]
[tex]\\ 6^{2} = 36[/tex]
[tex]\\ 6^{3} = 216[/tex]
[tex]\\ 6^{4} = 1296[/tex]
[tex]\\ 6^{5} = 7776[/tex]
One factor that is not defined in the question is whether the domain (the set of values for which a function is defined [WolframAlpha, 2019]) of the function [tex]\\ f(x) = 6^{x}[/tex] is only for x = {1, 2, 3, 4, 5} or defined for all possible values this function can take, that is, x = (-∞;∞) or all real numbers.
As you can see from the graph attached:
Theoretically, this function could take all values for x = (-∞;∞). If this were the case, the function is a continuous line with resulting values (that is, the range of the function) for f(x)>0 (0;∞). The graph below shows only positive values (x > 0) and values for x ≤ 5.For values x = {1, 2, 3, 4, 5}, the function is discrete (not a continuous line) and only defined for these values. In this case, the function would be represented only by the points in the graph, and the values it could take would be only f(x) = {6, 36, 216, 1296, 7776}, which would represent the range for this function.In the last case, we have to redefine the function as:
[tex]\\ f(x) = 6^{x}[/tex] for [tex] x = \{1, 2, 3, 4, 5\}[/tex].
What is the midpoint of 0.7 0.8?
Consider the functions f(x) = x2 − 13 and g(x) = x + 5. What is the value of (f − g)(−4)? ...?
Answer- The Answer is -12
Step-by-step explanation:
I took the test
Electrical energy is converted into which three forms of energy by a hair dryer?
A hair dryer converts electrical energy mainly into thermal energy to heat the air, but it also produces mechanical energy to power a fan and creates sound energy during operation.
A hair dryer converts electrical energy into three main forms of energy: thermal energy, mechanical energy, and sound energy. When a hair dryer is turned on, the electrical energy is primarily converted into thermal energy to heat the air. This is achieved by passing the electric current through resistive heating coils, which transfer energy to the surrounding air as heat. Furthermore, some of the electrical energy is transformed into mechanical energy to power a fan that blows the hot air out of the dryer. Lastly, the operation of the hair dryer also produces sound energy due to the movement of the fan and the rushing air.
a factory makes 1200 pounds of potato chips in four hours at this rate how many pounds of potato chips will a factory make in 10 hours
A factory producing 1200 pounds of chips in four hours will yield 3000 pounds when operating for 10 hours, assuming a consistent production rate.
Explanation:The given problem pertains to rate, which allows the determination of how much of something can be done in a specific amount of time. In this case, the rate of potato chip production is 1200 pounds every four hours. To find out how many pounds of potato chips can be produced in 10 hours, we first need to determine the hourly rate of production.
Firstly, we determine the hourly production rate of the factory by dividing the total produce by the time spent. That is 1200 pounds / 4 hours = 300 pounds/hour.
At this hourly rate, the amount of potato chips that can be produced in 10 hours can be calculated by multiplying the hourly rate by the number of hours, which is 300 pounds/hour * 10 hours = 3000 pounds.
So, a factory that produces 1200 pounds of potato chips in four hours will make 3000 pounds of potato chips in 10 hours.
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An 18 m tall tree is broken during a severe storm. The distance from the base of the trunk to the point where the tip touches the ground is 12 m. At what height did the tree break?
"If a parallelogram has four right angles, then it is a rectangle." What is the inverse of this conditional statement? A. If a parallelogram has four right angles, then it is a rectangle. B. If a parallelogram does not have four right angles, then it is a rectangle. C. If a parallelogram has four right angles, then it is not a rectangle. D. If a parallelogram does not have four right angles, then it is not a rectangle.
Answer:
If a parallelogram does not have four right angles, then it is not a rectangle.
Step-by-step explanation:
"If a parallelogram has four right angles, then it is a rectangle."
To get the inverse of the conditional statement, we take the negation of both the hypothesis and the conclusion.
So, the inverse of the given statement will be :
If a parallelogram does not have four right angles, then it is not a rectangle.
Which number has a cube root between 7 and 8?
A.57
B.153
C.244
D.499
8 foot piece of wood cut into how many 7/8 inches?
The table contains data on the mileage displayed on a bicycle odometer as it relates to the number of days since the bicycle was purchased. Using the Scatter Plot tool, find the value of the correlation coefficient for this data set.
0.8
1.0
0.3
0.1
Answer:
Hence, the correlation coefficient is:
r=1.0
Step-by-step explanation:
We know that the correlation coefficient ''r'' is calculated by the formula:
[tex]r=\dfrac{\sum{XY}}{\sqrt{\sum{X^2}\sum{Y^2}}}--------------(1)[/tex]
Let x denote the data point (Number of days since purchase)
Let y denote the data point (Mileage Displayed on odometer)
x y X=x-x' Y=y-y' XY X² Y²
15 67 -20 -95 1900 400 9025
25 122 -10 -40 400 100 1600
35 164 0 2 0 0 4
45 210 10 48 480 100 2304
55 247 20 85 1700 400 7225
x' be the mean of the data of the x-values.
[tex]x'=\dfrac{15+25+35+45+55}{5}\\\\\\x'=\dfrac{175}{5}\\\\\\x'=35[/tex]
Also let y' denote the mean of the y-values.
[tex]y'=\dfrac{67+122+164+210+247}{5}\\\\\\y'=\dfrac{810}{5}\\\\\\y'=162[/tex]
Now we have:
∑ XY=4480
∑ X²=1000
∑ Y²=20158
Hence, we put all the values in the formula (1) to obtain:
r=0.997 which is close to 1.0
Also from the scatter plot we could observe that the relationship is linear and also strong positive relationship.
Hence, the correlation coefficient is 1.0
How do you factorise xy zy?
The following data values represent a population. What is the variance of the values?
8, 10, 14, 12
A.22
B.11
C.5
D.10
Answer:
It’s actually 5
Step-by-step explanation:
Ape x
What is the first step to writing an equation of a line when given two points?
The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.
True
False
Answer:
true
Step-by-step explanation:
The statement 'The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.' is True.
What s graph of a function?"It is a set of points that satisfy given function."
What is transformation of a function?"It means that the curve representing the graph either 'moves to left / right/ up/ down' or 'it expands or compresses' or 'it reflects'."
What is translation of function?"A shift of the graph of a function up, down, left, or right, without changing the shape, size, or dimensions of the graph."
For given question,
We have been given a function y = 6cos x - 3
Also we have been given a transformed function y = 6cos(x - 2) - 3
We know that when function f(x) is transformed to f(x - k) then the graph of f(x) is translated 'k' units right.
So, the graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.
Therefore, the statement 'The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.' is True.
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the sum of two numbers is 85. twice one number plus four times the other is 218. find the numbers
The formula for an object's final velocity is f=i−gt f=i−gt . Solve for t.
answer: (f-i)/-g=t
I hope it helps~