a.) Consider the function f(x)=x^3−5x on the closed interval [−1,3]Find the exact value of the slope of the secant line connecting (−1 ,f ( −1)) and ( 3, f( 3))By the Mean Value Theorem, there exists c in (-1,3) so that m=f'(c). Find all values of such cc in (−1,3). Enter exact values. If there is more than one solution, separate them by a comma.m =c =
Answers
The exact values are m = 2 and c = √(7/3).
First, let's find the coordinates of the points (−1, f(−1)) and (3, f(3)):
f(x) = x^3 - 5x
f(-1) = (-1)^3 - 5(-1) = -1 + 5 = 4
f(3) = (3)^3 - 5(3) = 27 - 15 = 12
So, the points are (-1, 4) and (3, 12).
Next, let's find the slope of the secant line connecting these points:
m = (f(3) - f(-1))/(3 - (-1)) = (12 - 4)/(3 + 1) = 8/4 = 2
Now, by the Mean Value Theorem, there exists a value c in (-1, 3) such that m = f'(c). To find f'(x), we differentiate f(x):
f'(x) = d(x^3 - 5x)/dx = 3x^2 - 5
We found the slope of the secant line to be m = 2, so we set f'(c) = 2:
2 = 3c^2 - 5
To find the values of c, we solve this equation:
3c^2 - 5 = 2
3c^2 = 7
c^2 = 7/3
c = ±√(7/3)
Since c must be in the interval (-1, 3), we can discard the negative solution and keep the positive one:
c = √(7/3)
So the exact values are m = 2 and c = √(7/3).
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Related Questions
Please help 100 points
Answers
Answer:
Step-by-step explanation:
polydactyly is a fairly common congenital abnormality in which a baby is born with one or more extra fingers or toes. it is reported in about one child in every 500 . a young obstetrician celebrates her first 100 deliveries. assuming that these 100 births are unrelated and independent, what is the probability that the obstetrician has delivered no child with polydactyly? (enter your answer rounded to four decimal places, for example, 0.1111.)
Answers
The probability that the obstetrician has delivered no child with polydactyly is 0.8187.
To find the probability that the obstetrician has delivered no child with polydactyly, we can use the formula:
P(no polydactyly) = \((1 - P(polydactyly))^{number of births}\)
First, we need to find the probability of a child having polydactyly. This is given as 1 in every 500, which can be expressed as a decimal: 1/500 = 0.002.
Next, we find the probability of a child not having polydactyly: 1 - 0.002 = 0.998.
Now, we can find the probability of the obstetrician delivering no child with polydactyly in 100 unrelated and independent births by raising the probability of no polydactyly to the power of the number of births (100):
P(no polydactyly in 100 births) = (0.998)¹⁰⁰ ≈ 0.8187
So, the probability that the obstetrician has delivered no child with polydactyly is approximately 0.8187.
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a line passes through the points (-7, -7) and (7, -5) what is this equation in slope intercept form
Answers
Answer:
\(y = \frac{6}{7}x - 1\)
Explanation:
\(Point 1: (-7, -7)\\Point 2: (7, -5)\)
\(\frac{y_{2} - y_{1} }{x_{2} - x_{1}}\)
\(\frac{(-5) - (-7)}{(-7) - (7)}\)
\(\frac{-12}{-14}\)
\(\frac{12}{14}\)
\(m = \frac{6}{7}\)
\(y - y_{1} = m(x - x_{1})\)
\(y - (-7) = \frac{6}{7}(x - (-7))\)
\(y + 7 = \frac{6}{7}(x + 7)\)
\(y = \frac{6}{7}x + \frac{42}{7} - 7\)
\(y = \frac{6}{7}x + (\frac{42}{7} - \frac{49}{7})\)
\(y = \frac{6}{7}x - 1\)
Help anyone????????????
Answers
Answer:
A new car would cost 400$
If 140$ is 35% of the price of a new car then all we need to do is find how much 1% of that new car is
35%:140$
divide by 35 on each side
1%:4$
If 1% is 4$ then 100% is 400$
Hope this helped
Step-by-step explanation:
Which step in the construction of copying a line segment ensures that the new line segment has
the same length as the original line segment?
Answers
Answer:
希 · 斯杰德布德克克贾克什伊索伊菲夫杰杰德克斯克克沃斯克维约沃沃
Answer:
Now out of these points, i guess steps 5 and 6 are the main steps that ensure that the copied line segment is exactly the same as the original segment.
don't know
If 8 people consisting of 4 couples are randomly arranged in a row, find the probability that no person is next to their partner.
Answers
The Probability that no person is next to their partner is 104/105.
Probability of an event E represented by P(E) can be defined as (The number of favorable outcomes )/(Total number of outcomes).
Permutations is defined as arrangement of elements/objects in a particular way.
According to the question ,
8 people can be arranged in 8! ways = 40320ways
First let us find the probability that person is next to their partner.
4 couples can be arranged in 4! ways
and the couple itself can be arranged in 2! ways
Since there are 4 couples ,
Number of arrangement = \(4!(2!)^{4}\)
=24x16
=384 ways
Probability that person is next to their partner = \(\frac{384}{40320} =\frac{1}{105}\)
and the probability that no person is next to their partner is \(1-\frac{1}{105} =\frac{104}{105}\).
Therefore , the probability that no person is next to their partner is \(\frac{104}{105}\).
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water is leaking out of an inverted conical tank at a rate of 10,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. the tank has height 6 m and the diameter at the top is 4 m. if the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate (in cm3/min) at which water is being pumped into the tank.
Answers
The rate at which water is being pumped into the tank is 11,760 cm3/min.
How to find rate?The formula for the volume of a conical tank is:V = (1/3)πr2h
Where r is the radius of the tank, and h is the height of the tank.Find the radius of the tank at a height of 2 m.Using similar triangles:
(r / 2) = (2 / 6)
r = 2/3 * 4r = 8/3 cm
The formula for the rate of change of volume of a conical tank is:dV / dt = (πr2 / 3)dh / dt
dV / dt = pump rate - leak rate
= pump rate - 10,500 cm3/mindh / dt
= 20 cm/minr = 8/3 cm
Plug in the values:pump rate - 10,500 = (π(8/3)2 / 3) * 20pump rate - 10,500
= 2114.67pump rate
= 11,760 cm3/min
Therefore, the rate at which water is being pumped into the tank is 11,760 cm3/min.
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find the missing angles
Answers
Answer:
1: 41
2: 85
3: 95
4: 85
5: 36
6: 49
7: 106
I’ll give brainliest
Completed explication for polynomials with examples
Answers
Answer:
am algebraic expression made up of a collection of terms, typically separated by a + or -
*a term that can be made up of exponents, coefficients, & variables
example:
4x + 2x - 6
Find the optimal values of x and y using the graphical solution method: Min x + y subject to: x + y ≥ 7 5x + 2y ≥ 20 x ≥ 0, y ≥ 0.
Answers
The optimal values of x and y that minimize the objective-function x + y, subject to the given constraints, are x = 4 and y = 0.
We can find the corner points of the feasible region and evaluate the objective function at those points to determine the optimal solution.
Graph the constraints:
Start by graphing the inequalities:
x + y ≥ 7
5x + 2y ≥ 20
x ≥ 0
y ≥ 0
Plot the lines x + y = 7 and 5x + 2y = 20. To graph x + y = 7, plot two points that satisfy the equation, such as (0, 7) and (7, 0), and draw a line through them. To graph 5x + 2y = 20, plot two points such as (0, 10) and (4, 0), and draw a line through them.
Shade the region that satisfies the inequalities x ≥ 0 and y ≥ 0.
The feasible region will be the shaded region.
Identify the feasible region:
The feasible region is the shaded region where all the constraints are satisfied. In this case, the feasible region will be a polygon bounded by the lines x + y = 7, 5x + 2y = 20, x = 0, and y = 0.
Find the corner points:
Locate the intersection points of the lines and the axes within the feasible region. These are the corner points. In this case, we have the following corner points:
Intersection of x + y = 7 and x = 0: (0, 7)
Intersection of x + y = 7 and y = 0: (7, 0)
Intersection of 5x + 2y = 20 and x = 0: (0, 10)
Intersection of 5x + 2y = 20 and y = 0: (4, 0)
Evaluate the objective function:
Evaluate the objective function, which is x + y, at each corner point:
(0, 7): x + y = 0 + 7 = 7
(7, 0): x + y = 7 + 0 = 7
(0, 10): x + y = 0 + 10 = 10
(4, 0): x + y = 4 + 0 = 4
Determine the optimal solution:
The optimal solution is the corner point that minimizes the objective function (x + y). In this case, the optimal solution is (4, 0) because it has the smallest objective function value of 4.
Therefore, the optimal values of x and y that minimize the objective function x + y, subject to the given constraints, are x = 4 and y = 0.
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Andrew wanted to know the circumference of his bicycle tires he measured the radius which is 13 inches what is the circumference of the tire
Answers
Answer:
81.64 inches
Step-by-step explanation:
The equation to find the circumference is
C = 2(3.14) (r)
C = 2(3.14) (13)
C = 81.64 inches
So, the circumference of the tire is 81.64 inches
5617 rounded to the nearest thousand
Answers
Answer:
6000
Step-by-step explanation:
the rule for rounding is if its 4 and below round down, and 5 and up its round up. so going by the rule, you would see that the hundred place is 6 and so round it up to 6000! plz mark as brainliest :D
Jason is playing a trivia game with his friends. At the end of each round, his score updates to the square of 1 less than the previous round’s score. If he has 8 points in the first round, which of the following recursive formulas can be used to determine his score at the end of future rounds? Assume that the number of rounds is unlimited, and n is the number of rounds.
Answers
Answer:
8+n^2
Step-by-step explanation:
Answer:
f(n+1)=(f[n]-1)^2, where f(1)=8
Step-by-step explanation:
"it appears your explanation includes a link" so you don't get an explanation
please help! i need help with this problem. i have the answer but i have no idea how to factor it
Answers
Answer:
(q²-p)(p²-q), 240
Step-by-step explanation:
Q1 p²q²+pq-q³-p³=(p²q²-q³)+(pq-p³)
=q²(p²-q)-(p³-pq)
=q²(p²-q)-p(p²-q)
=(q²-p)(p²-q)
Q2 If p=4 and q=-4
(4)²(-4)²+(4) x(-4)-(-4)³-4³=256-16+64-64=240
Answer:
please mark as brainliest
Find the arc length along a circle of radius 10 units subtended by an angle of 275°
Enter the exact answer.
Answers
The arc length along a circle with a radius of 10 units and a subtended central angle of 275° is 27.5π units.
To find the arc length (s) along a circle, we use the formula:
s = rθ
Given that the radius (r) is 10 units and the central angle (θ) is 275°, we need to convert the angle to radians.
θ (in radians) = θ (in degrees) * π/180
θ = 275° * π/180
θ = (11π/4) radians
Now we can substitute the values into the formula to calculate the arc length:
s = rθ
s = 10 * (11π/4)
s = (110π/4)
s = 27.5π
Therefore, the exact answer for the arc length is 27.5π units.
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quantity demanded pairs of shoes quantity supplied \( \quad \) pairs of shoes Will there be a surplus or shortfall at this price? There will be a surplus. There will be a shortfall.
Answers
The quantity demanded is greater the quantity supplied, and as such there will be a shortfall.
How to solve demand and supply functions?The demand and supply functions are given as:
Demand: 2p + 5q = 200
Supply: p - 29 - 30
For a price of p = $70, we will have:
Demand: 2(70) + 5q = 200
140 + 5q = 200
5q = 60
q = 12
Supply: 70 - 29 - 30 = 11
Therefore, at a price of $70, the quantity demanded is 12 pairs of shoes and the quantity supplied is 11 pairs of shoes.
Since the quantity demanded is greater the quantity supplied, then there will be a shortfall of 1 pair of shoes.
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Complete question is:
If the demand for a pair of shoes is given by 2p + 5q = 200 and the supply function for it is p - 29 - 30, compare the quantity demanded and the quantity supplied when the price is $70. quantity demanded pairs of shoes quantity supplied pairs of shoes Will there be a surplus or shortfall at this price? There will be a surplus. There will be a shortfall.
An electron (charge−1.6× 10−19 c) moves on a path perpendicular to the direction of a uniform electric field of strength 2.0 n/c. how much work is done on the electron as it moves 14.0 cm?
Answers
I.e E=F/C
F=E x C
F= 2 x -1.6 x10^-19 C
F= -3.2 x 10^-19 N
Work done = F x D
= -3.2 x 10^-19 N x 14/100
=0.448 x 10^-19
Note:use of x is sign of multiplication
(d) water is pumped into the tank. when the height of the water is 5 feet, the height is increasing at the rate of 0.26 feet per minute. using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. indicate units of measure.
please hurry!!! timed quiz!!!!!!
formula is a(y)= bounds of y f(x) dx
Answers
The rate at which the volume of water is changing with respect to time when the height of the water is 5 feet is (5.2/3)π√3 cubic feet per minute.
When the height is 5 feet, it is stated that the height of the water in a tank rises at a pace of 0.26 feet per minute.
When the water reaches five feet high, we can determine the rate of change in water volume with respect to time using the model from part (c).
We may calculate the volume of water in the tank using the formula V = (1/3)r2h, where r is equal to 3 feet and h is the height of the water in feet.
The Pythagorean theorem can be used to get the tank's radius at a height of 5 feet: r = ((102 - 52) = 75 = 53 feet.
Taking the derivative of the volume with respect to time, we get:
dV/dt = (1/3)π(2r)(dh/dt)
Substituting the values we have:
dV/dt = (1/3)π(2(5√3))(0.26)
= (5.2/3)π√3 cubic feet per minute
As a result, when the water level is 5 feet high, the rate of change in water volume with respect to time is (5.2/3)3 cubic feet per minute.
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A red, blue, and green die are thrown. Each die has six possible outcomes. How many outcomes are possible in which the three dice all show different numbers
Answers
The total number of outcomes where all three dice show different numbers is 120.
To determine the number of outcomes in which all three dice show different numbers, you can use the multiplication principle. First, choose a number on the red die (6 options). Then, choose a different number on the blue die (5 options, since it cannot be the same as the red die). Finally, choose a different number on the green die (4 options, since it cannot be the same as the red or blue die). Multiply the options together: 6 x 5 x 4 = 120. Therefore, there are 120 possible outcomes in which the three dice all show different numbers.
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Can you help me please..!
Answers
Чx-7=-3x solve the expression
Answers
Answer:
x=1
Step-by-step explanation:
Answer:
x=1
Step-by-step explanation:
4x−7=−3x
Step 1: Add 3x to both sides.
4x−7+3x=−3x+3x
7x−7=0
Step 2: Add 7 to both sides.
7x−7+7=0+7
7x=7
Step 3: Divide both sides by 7.
7x
7
=
7
7
x=1
Melissa just finished a new book about a time-traveling magician. She read the same amount every day and completed the book in just 10 days! The book had 85 pages.
How many pages did Melissa read each day?
Write your answer as a proper fraction or mixed number.
Answers
Answer:
8 1/2
Step-by-step explanation:
She read 8 1/2 pages each day because if you divide 85 by 10 you will get 8 1/2.
What does thirty five plus negative ten equal
Answers
Hope i helped!
Answer:
Step-by-step explanation:
35 plus negative 10 equals 25. Basically, the simple way to say it is 35 minus 10 equals 25.
Find the missing side length.
Answers
Answer:
13
Step-by-step explanation:
The top must be the same length as the total of the bottom
top = 5+8
top = 13
What is the percent change from 147 to 298? Round your answer to the nearest tenth.
Answers
Answer:
(147 - 298)/298 x 100
-151/298 × 100
Change ≈ -50.6711%
Step-by-step explanation:
Answer:
102.7
Step-by-step explanation:
your starting number is 147,
your earned number is 298,
to get the percentage, it will be in decimal form
the answer is-
102.72108843537416 %
Now you round to the nearest tenth-
102.7
I hope this helps you-
Step 1: Calculate the change (subtract old value from the new value) ...
Step 2: Divide that change by the old value (you will get a decimal number)
Step 3: Convert that to a percentage (by multiplying by 100 and adding a "%" sign)
Select the two values of x that are roots of this equation.
x^2+3x-5=0
Answers
\( \: \)
\({ \rm{ \bold{C )x = \frac{ - 3 - \sqrt{29} }{2} }}}\)\( \: \)
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Given:-\( \rm {x}^{2} + 3x - 5 = 0\)\( \: \)
By using quadratic equation formula:-
\( \rm \bold{{a {x}^{2} + bx + c = 0 }}\)\( \: \)
Formula:-\( \boxed{ \rm{ \red{x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} }}}\)\( \: \)
Solution:-\(\boxed{ \underline{ \rm \bold{\: a = 1, b = 3 , c = -5 }}}\)\( \: \)
\( \rm{ \bold{x = \frac{ - b \pm\sqrt{ {b }^{2} - 4ac } }{2a}} }\)\( \: \)
\( \rm \: x \: \frac{ - 3 + \sqrt{ {(3)}^{2} - 4 \times 1 \times ( - 5) } }{2 \times 1} \)\( \: \)
\( \rm \: x \: \frac{ - 3 + \sqrt{ 9- 4 \times ( - 5) } }{2 } \)\( \: \)
\( \rm \:x = \frac{ - 3 + \sqrt{9 - ( - 20)} }{2} \)\( \: \)
\( \underline{\boxed{ \green{ \rm \bold{ \: x = \frac{ - 3 + \sqrt{29} }{2} }}}}\)\( \: \)
and ,
\( \rm \: x \: \frac{ - 3 - \sqrt{ {(3)}^{2} - 4 \times 1 \times ( - 5) } }{2 \times 1} \)\( \: \)
\( \rm \: x \: \frac{ - 3 - \sqrt{ 9- 4 \times ( - 5) } }{2 } \)\( \: \)
\( \rm{x = \frac{ - 3 - \sqrt{9 - ( - 20)} }{2} }\)\( \: \)
\( \underline{ \boxed{ \rm{ \bold{\color{green}x = \frac{ - 3 - \sqrt{29} }{2} }}}}\)\( \: \)
hope it helps! :)
Two joggers run 6 miles south and then 5 miles east. What is the shortestdistance they must travel to return to their starting point?
Answers
Answer:
7.81 miles
Step-by-step explanation:
pythagorean theorem, 6 units downwards, and 5 east, so we have to calculate the hypotenuse, or sqrt( 6^2 + 5^2) which is sqrt61 or 7.81 miles
A customer in a coffee shop purchases a blend of two coffees: Kenyan, costing $3.50 a pound, and Sri Lankan, costing $5.60 a pound. He buys 3 lb of the blend, which costs him $11.55. How many pounds of each kind went into the mixture
Answers
Answer:
10.55.
Step-by-step explanation:
no
The solid below is dilated by a scale factor of . Find the surface area of the solid
created upon dilation. Answer in terms of t.
9
9
Answers
The surface area of the cylinder after dilation is: 36 units².
What is the surface Area of a Cylinder?Surface area of a cylinder = 2πrh + 2πr²
Find the surface area of the cylinder before dilation:
Surface area = 2πrh + 2πr² = 2(π)(9)(9) + 2(π)(9²)
Surface area = 324π units²
Surface area of the cylinder after dilation = 324π × (1/3)²
= 324π × 1/9
= 36 units²
Thus, the surface area of the cylinder after dilation is: 36 units².
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in a barn with cows and chickens, the number of legs was 16 more than twice the number of heads. how many cows were in the barn?
Answers
The total number of cows in the barn is 7 and the number of cows should be a whole number on substituting different values of h.
Let the number of cows be "c" and the number of chickens be "h".
We know that cows have 4 legs and chickens have 2 legs.
In this case, the total number of legs would be 4c + 2h.
The total number of heads would be c + h.
According to the problem, the number of legs is 16 more than twice the number of heads.
Mathematically, this can be represented as:
4c + 2h = 2(c + h) + 16
Simplifying this equation, we get:
2c - 2h = 8c - 16
hence, 6c = 14h or 3c = 7h.
Therefore, the number of cows in the barn can be calculated as:
c = (7/3)h
Since the number of cows should be a whole number, we can substitute different values of h and see which one gives a whole number answer for c.
For example, if h = 3, then:
c = (7/3) x 3 = 7, which is a whole number.
So, there are 7 cows in the barn.
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