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) =
Answers
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]
Related Questions
Give 9.7814 correct to the nearest whole number
Answers
because you're rounding
Final answer:
9.7814 rounded to the nearest whole number is 10, as the first digit after the decimal point is 7, which requires rounding up.
Explanation:
To find the nearest whole number to 9.7814, we need to look at the first digit after the decimal point. Since this digit is 7 (which is equal to or greater than 5), we round up the number. Rounding up changes 9.7814 to the next whole number, which is 10.
The general rule for rounding to the nearest whole number is:
If the first decimal place is 5 or greater, round up.If the first decimal place is less than 5, round down.Therefore, 9.7814 rounded to the nearest whole number is 10.
Dayla earns $108.75 for working 15 hours of holiday open rap at this rate how much money will see you earn if she works 18 hours the next week explain
Answers
15 hours to $108.75
1 hour to 108.75/15 = $7.25 per hour
dayla would earn (7.25*18) for the preceding week which $130.50
A store is having a 10% off sale. It gives an additional 20% off for an item that is above $100 after the first 10% discount.A jacket costs $124.99
if you buy the jacket, how much will you save in total?
*SHOW WORK* ...?
Answers
To calculate the total amount saved when buying a jacket at a store with a 10% discount and an additional 20% discount, follow these steps: find the savings from the first discount, calculate the price after the first discount, check if the price is above $100, calculate the additional discount, and add both savings together. In this case, the total amount saved is $34.9972.
Explanation:To calculate the total amount saved, we need to use the following steps:
Find the total savings from the first 10% discount: $124.99 * 0.10 = $12.499Find the price of the jacket after the first discount: $124.99 - $12.499 = $112.491Check if the price is above $100. If it is, calculate the additional 20% discount: $112.491 * 0.20 = $22.4982Add the savings from the additional discount to the previous savings: $12.499 + $22.4982 = $34.9972So, the total amount saved is $34.9972.
Final answer:
To find the total amount saved, apply the discounts and add the amounts together. The total amount saved is $35.00.
Explanation:
To find the total amount saved, you need to calculate the discount for each discount percentage. First, apply the 10% discount to the original price of the jacket:
10% of $124.99 = $12.499.
Next, apply the 20% discount to the price after the first discount:
20% of ($124.99 - $12.499) = 20% of $112.491 = $22.4982.
Finally, add the two discounts together to find the total amount saved:
$12.499 + $22.4982 = $34.9972.
Rounding to the nearest cent, the total amount saved is $35.00.
Which of the following is the statement of the triangle inequality theorem?
A. The sum of the measures of any two angles of a triangle is greater than the measure of the third angle.
B. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
C. The longest side of a triangle is opposite the largest angle.
D. The largest angle of a triangle is between the two longest sides.
Answers
we know that
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side
so
Let
a,b,c------> the length sides of a triangle
The theorem states that three conditions must be met
case 1)
[tex] a+b > c [/tex]
case 2)
[tex] a+c > b [/tex]
case3)
[tex] c+b > a [/tex]
therefore
the answer is the option
B. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
what is 2/3x5/6x14 can you help me on this problem
Answers
Answer: 7 wholes and 63/81
what are the translation rules in geometry?
Answers
Final answer:
The translation rules in geometry involve moving every point in a figure the same distance and direction. For a horizontal translation, add or subtract the same value from the x-coordinate of each point. For a vertical translation, add or subtract the same value from the y-coordinate of each point.
Explanation:
In geometry, translation is a transformation that moves every point in a figure the same distance and direction. The translation rules involve determining the new coordinates of the translated figure.
The translation rules in geometry can be summarized as:
For a horizontal translation, add or subtract the same value from the x-coordinate of each point.For a vertical translation, add or subtract the same value from the y-coordinate of each point.For example, if you have a point with coordinates (2, 3) and you want to translate it 3 units to the right and 2 units up, you would add 3 to the x-coordinate and 2 to the y-coordinate to get the new coordinates of the translated point: (5, 5).
What is the ratio of julios 22 to jonah 8 points?
Answers
the answer is 11:4
The hand on a clock shows 6:00pm what is the measure of the angle
Answers
Explain what the m and b variables represent in the y=mx b equation
Answers
A bag cost 10.50 but you get 20 percent off how much is the bag now?
Answers
Which is
$10.50×.2=$8.40
How many millimeters are in 50 centimeters?
Answers
so, 50x10=500
13 tenths plus 8 tenths plus 32 hundredths <,> =,to 2.42
Answers
What is the solution to the system of equations?
y = 1.5x – 4
y = –x
a. (–1.6, 1.6)
b. (–1.5, 1.5)
c. (1.5, –1.5)
d. (1.6, –1.6)
Answers
y= 1.5x - 4
-x= 1.5x - 4
1.5x+x=4
2.5x=4
x=1.6
y= -1.6
What is the square root of 59,005?
Answers
A sneaker store salesman had $4,125 in total monthly sales last month. He made $165 in commission from those sales. What is the salesman's commission as a percent of his total monthly sales? Enter your answer in the box. %
Answers
Answer:
Salesman's commission 4% of total monthly sales.
Step-by-step explanation:
A sneaker store salesman had $4,125 in total monthly sales last month.
He made $165 in commission from those sales.
Suppose, it is x% of the total sale.
So, x% of $4125 will be,
[tex]=4125\times \dfrac{x}{100}[/tex]
According to the question it is equal to $165, so
[tex]\Rightarrow 4125\times \dfrac{x}{100}=165[/tex]
[tex]\Rightarrow x=\dfrac{165\times 100}{4125}[/tex]
[tex]\Rightarrow x=\dfrac{16500}{4125}[/tex]
[tex]\Rightarrow x=4\%[/tex]
Therefore, salesman's commission 4% of total monthly sales.
Answer: 4%
Step-by-step explanation:
Due to the fact that the salesman had $4,125 in total monthly sales the previous month, the answer would be 4%
graph the equation. y = 4x - 3
Answers
y times 6 for y = 2/3
Answers
6y = 2/3
y = 2/3*6
y = 2/18
y = 1/9
So, your final answer is 1/9
Hope this helps!
The width of a square-based storage tank is 3m less than its height. The tank has a capacity of 20 m3. If the dimensions are integer values in metres, what are they?
Answers
The volume of a rectangle is: V = w * l * h (w - width, l - length, h - height)
A rectangle is square based, so: w = l
V = w * w * h = w² * h
The width of a square-based storage tank is 3m less than its height: w = h - 3
V = 20 m³
w² * h = 20
(h - 3)² * h = 20
(h² - 2 * h * 3 + 3²) * h = 20
(h² - 6h + 9) * h = 20
h² * h - 6h * h + 9 * h = 20
h³ - 6h² + 9h = 20
h³ - 6h² + 9h - 20 = 0
h * h² - 5h² - h² + 5h + 4h - 4*5 =0
(h * h² - 5 * h²) - (h * h - 5h) + (4*h - 4*5) =0
h²(h - 5) - h(h - 5) + 4(h - 5) = 0
(h - 5)(h² - h + 4) = 0
We have that one solution is:
h - 5 = 0
h1 = 5
Other two solutions are:
h² - h + 4 = 0
Quadratic equation is: ax² + bx + c = 0
[tex]h_{2,3} = \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} = \frac{-(-1)+/- \sqrt{ (-1)^{2}-4*1*4 } }{2*1} = \frac{1+/- \sqrt{1-16} }{2} = \\ \\ = \frac{1+/- \sqrt{-15} }{2} = \frac{1+/- \sqrt{-1*15} }{2} = \frac{1+/- \sqrt{-1} * \sqrt{15} }{2}= \frac{1+/- i\sqrt{15} }{2} \\ \\ h_2 = \frac{1+ i\sqrt{15} }{2} \\ \\ h_3 = \frac{1- i\sqrt{15} }{2}[/tex]
The only integer value is h1 = 5 m
h = 5 m
w = h - 3 = 5 - 3 = 2 m
Volume = x(x)(x + 3) = x^3 + 3x^2 = 20
x^3 + 3x^2 - 20 = 0
x = 2
Therefore, the dimension of the tank is 2 m by 2 m by 5 m
What number should be added to the expression x^2 + 2x to change it into a perfect square trinomial?
Answers
so x^2 + 2x must be added 1 to become x²+2x+1=(x+1)²
Solve for x and show work. 2xln(1/x)-x=0
Answers
Then divide 2x and it will be
In(1/x)=1/2
Then
e^1/2=1/x
1.65=1/x
x=1/1.65
Assume the function has an inverse. Without solving for the inverse find the indicated function values: f(x)=x^3+4x-1, (a) f^-1 (-1) (b) f^-1 (4) ...?
Answers
To find the indicated function values of the inverse of a function, substitute the given values into the original function and solve for x.
Explanation:To find the indicated function values of the inverse of the function f(x)=x^3+4x-1, without solving for the inverse function, we substitute the given values into the original function and solve for x.
(a) To find f^-1(-1), we substitute -1 into the function: f(-1)=(-1)^3+4(-1)-1 = -1+(-4)-1 = -6.
(b) To find f^-1(4), we substitute 4 into the function: f(4)=(4)^3+4(4)-1 = 64+16-1 = 79.
If f(x) is an odd function and the graph of f(x) includes points in Quadrant IV, which statement about the graph of f(x) must be true?
It includes points in Quadrant I.
It includes points in Quadrant II.
It does not include points in Quadrant I.
It does not include points in Quadrant II.
Answers
Answer:
It includes points in Quadrant II.
It does not include points in Quadrant I.
Step-by-step explanation:
Given : If f(x) is an odd function and the graph of f(x) includes points in Quadrant IV.
To find : which statement about the graph of f(x) must be true.
Solution : We have given that f(x) is an odd function is lies in Quadrant IV.
Definition of an odd function: the y-value of the function at negative x is always opposite sign as the y-value of function at opposite x.
Like (x ,-y) or (x , -y)
In odd function x and y values are in opposite sign that mean odd function always lies in Quadrant II and Quadrant IV.
It does not include points in Quadrant I
Therefore, It includes points in Quadrant II.
It does not include points in Quadrant I.
Answer:
includes quad 2
Step-by-step explanation:
Which equation shows the substitution method being used to solve the system of linear equations?
x – y = 7
x = y + 3
A.
(y + 3) – y = 7
B.
x = (7 – x) + 3
C.
x – (y + 3) = 7
D.
x – y = y + 3
Answers
Read the numbers and decide what the next number should be 8,4,2,6,3,2,4.
Answers
The given sequence involves a two-part pattern. One part is halving the previous number and the other part alternates between 4 and 6. Considering the pattern, the next number should be 0.5.
Explanation:The sequence 8,4,2,6,3,2,4 appears to have a pattern. It seems to be a two-part pattern that alternates. The first part, observed in the numbers 8, 2, and 2, is halving (8 divided by 2 is 4, and 2 divided by 2 is 1, and 1 by 2 is 0.5). Then, the second part, observed in 4,6,4, seems to alternate between 4 and 6. So, if the pattern continues, the next number after 4 should be 0.5 from the first part of the pattern.
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The sequence could actually be representing two series - one series is halving the number (8 to 4 to 2) and the other is multiplying each number by 2 (3 to 2 to 4 to 8). Thus, the next number could either be 1 (if we continue the halving series) or 12 (if we continue the multiplying by 2 series).
Explanation:The number sequence provided seems to be made of two different series interwoven. One series is halving each number (8 to 4 to 2), and the other series is multiplying each number by 2 (2 to 4 to 6 to 12).
Therefore, the next number after 8, 4, 2, 6, 3, 2, 4 should be 1 (half of 2) whereas the next number in the second series after 6, 3, 2, 4 should be 12 (four times 3). So, depending on which pattern the sequence is following, the next number could be either 1 or 12.
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A conical tank has height 3 m and radius 2 m at the top. Water flows in at a rate of 0.7m^3/min. How fast is the water level rising when it is 0.8 m?
Answers
look at the image
r / 2 = h / 3 r = 2/3h V = 1/3(pi)r^2h . Put r = 2/3h V = 1/3(pi)(4/9)h^3 = 4(pi)/27h^3 dV/dt = 4(pi)/27(3h^2)dh/dt = 4/9(pi)(h^2)dh/dt Plug in the numbers and solve for dh/dt at h = 0.8
dV/dt = 4/9(pi)(h^2)dh/dt = 1.396h^2dh/dt dV/dt = 0.7, h = 0.8 0.7 = 1.39(.8^2)dh/dt dh/dt = 0.78 meters / min
If h(x) = -3x - 10, find h(-3).
Answer
-1
2.3
-19
1
...?
Answers
__________________________________________________________
Explanation:
__________________________________________________________
Given: " If h(x) = -3x − 10, find h(-3). "
So; we plug in "-3" for all values of "x" ; to solve for " h(-3)" ; as follows:
→ h(-3) = -3(-3) − 10 = 9 − 10 = - 1 .
The answer is: " - 1 " .
__________________________________________________________
Linda Frankell pays $50.00 a month for group health insurance. There is a $300 deductible. Her medical bills for the last year were $2,600.00. Linda's insurance company provided payment of 80% of the bills less the deductible.
What was the insurance company's payment? $
What was Linda's total cost (including the monthly premium)? $
Answers
Linda's premiums for 12 months = $50 × 12 = $600
Deductible is the amount Linda Frankell will have to pay each year as a policy holder towards her medical expenses before the insurance company pays its share.
Amount Linda pays towards deductible = $300
Linda's medical bills = $2,600.00
The insurance company will pay 80% of the bills less the deductible, which means, the insurance company will pay 80% of the ($2,600.00 - $300 deductible = $2,300.00)
∴ Amount the insurance company pays = 80% of $2,300.00 = $1,840.00
Amount Linda has to pay towards the $2,300.00 = $2,300.00 - $1,840.00 = $460.00
Linda's total cost = Linda's 12 months premiums + $300 deductible + $460 = $600 + $300 + $460 = $1,360.00
Insurance company's payment = $1840.00
Linda's total cost = $1,360.00
Simplify: log base 3 (1/3) ...?
Answers
[tex]log_3( \frac{1}{3} )[/tex]
_____
Since [tex]log_y(x) = \frac{log_{10}(x)}{log_{10}(y)} [/tex], then: [tex]log_3( \frac{1}{3} )= \frac{log_{10} ( \frac{1}{3}) }{log_{10}(3)} [/tex]
_____
Since [tex] \frac{1}{x} = x^{-1} [/tex], then: [tex]\frac{log_{10} ( \frac{1}{3}) }{log_{10}(3)} = \frac{log_{10}(3^{-1} )}{log_{10}(3)} [/tex]
_____
Since [tex]log x^{a} =a*logx[/tex], then: [tex]\frac{log_{10}(3^{-1} )}{log_{10}(3)}= \frac{-1*log_{10}(3)}{log_{10}(3)} [/tex]
From here: [tex]\frac{-1*log_{10}(3)}{log_{10}(3)}=-1*\frac{log_{10}(3)}{log_{10}(3)}=-1*1 = -1[/tex]
what is the 10th term of this geometric sequence?
1,2,4 ,8...
A)128
B)256
C)512
D)1024
Answers
16 × 2 = 32 = 6th term
32 × 2 = 64 = 7th term
64 × 2 = 128 = 8th term
128 × 2 = 256 = 9th term
256 × 2 = 512 = 10th term
The answer is 512
Hope this helps!
What is the solution of the equation?
4p/6+27=39
A. 22
B. 99
C. 4
D. 18
Answers
We simplify the equation to the form, which is simple to understand
4p/6+27=39
Simplifying:
+ 0.666666666667p+27=39
We move all terms containing p to the left and all other terms to the right.
+ 0.666666666667p=+39-27
We simplify left and right side of the equation.
+ 0.666666666667p=+12
We divide both sides of the equation by 0.666666666667 to get p.
p=18