Answers
a. The value of f'(x) = 8x³.
b. The gradient of the graph of f at x = -1 is -8.
c. The point (x₁, y₁) is (-1, 3).
d. The equation of the normal line when x = -1 is y = (1/8)x + 25/8.
How to calculate the valuea. Using the power rule, we differentiate each term:
f'(x) = d/dx (2x⁴) + d/dx (1)
Differentiating 2x⁴
f'(x) = 2 * 4x³ + 0
= 8x³
b. To find the gradient of the graph of f at x = -1, we substitute x = -1 into the derivative we found in part a.
f'(-1) = 8(-1)³
= 8(-1)
= -8
The gradient of the graph of f at x = -1 is -8.
c. We already know that the gradient (m) is -8, and the point (x₁, y₁) is (-1, f(-1)). We can find f(-1) by substituting x = -1 into the original function:
f(-1) = 2(-1)⁴ + 1
= 2(1) + 1
= 2 + 1
= 3
So the point (x₁, y₁) is (-1, 3).
Using the point-slope form of the line equation, we can substitute the values:
y - 3 = -8(x - (-1))
y - 3 = -8(x + 1)
y - 3 = -8x - 8
y = -8x - 5
Therefore, the equation of the tangent line when x = -1 is y = -8x - 5.
d. The normal line is perpendicular to the tangent line. The gradient of the normal line is the negative reciprocal of the gradient of the tangent line.
The gradient of the tangent line we found in part c is -8. Therefore, the gradient of the normal line is -1/(-8), which simplifies to 1/8.
Using the point-slope form of the line equation, we substitute the gradient (1/8) and the point (x₁, y₁) = (-1, 3):
y - 3 = (1/8)(x - (-1))
y - 3 = (1/8)(x + 1)
y - 3 = (1/8)x + 1/8
y = (1/8)x + 25/8
Therefore, the equation of the normal line when x = -1 is y = (1/8)x + 25/8.
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Related Questions
Can someone please help me in this question
Answers
Step-by-step explanation:
a) Let y = f(x) = 3x - 2x^2
f(-2) = 3(-2) - 2(-2)^2 = p
= -6 - 8
= -14
= p
f(2.5) = 3(2.5) - 2(2.5)^2 = q
= 7.5 - 12.5
= -5
= q
b) graphing
c) From the graph, you should be able to verify the following:
i) f(0.5) = 3(0.5) - 2(0.5)^2 = 1
ii) 0.5 = 3x - 2x^2 or x = 1.3, 0.2
iii) the maximum occurs at
f(0.75) = 1.125
d) the equation for the line of symmetry is x = 0.75
2y=-5/6x+5
Write in standard form
Answers
Answer:
5x + 12y = 30
Step-by-step explanation:
The equation of a line in standard form is
Ax + By = C ( A is a positive integer and B, C are integers )
Given
2y = - \(\frac{5}{6}\) x + 5 ( multiply through by 6 to clear the fraction )
12y = - 5x + 30 ( add 5x to both sides )
5x + 12y = 30 ← in standard form
A farmstand sells apples, a, for $4 a bucket; peaches, p, for $6 A bucket; and strawberries, s, for $9 a bucket. The stand earn $229 in revenue last month. The stand sold half as many strawberry buckets as peach buckets, and eight were apple buckets then peach buckets. Using the substitution method how many apples peaches and strawberries did the farmstand sell?
A. a=10; p=18; s=9
B. a=4; p=6; s=9
C. a=18; p=10; s=9
D. a=9; p=18; s=10
Answers
Answer:
A.
Step-by-step explanation:
process of elimination. They sold half as many strawberries as peaches.
q - 0.63 = 1.16
solve for q. please
Answers
Answer: Q=1.79
Step-by-step explanation: Move 0.63 to the right and add both numbers then calculate it. So Q=1.79
Expand the following
4(×-3)
5(2p-4)
‐6(7×-4y
Answers
2) 10p-20
3) -42x + 24y
help me please !!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
Step-by-step explanation:
Pack value for 1 = 6/2.79 = 2.1
this is better as clearly, 2.1<.27
Although, Lara Croft was successful at stopping the enemy in
Tomb Raider, it was discovered that one artifact is still missing.
Luckily, the whereabouts are known: A cruise ship leaving
Seattle. By the time Ms. Croft arrives in Seattle, she has missed
the boat by 5 hours. What should she do?
FACTS:
- The cruise ship can go 528 mi in one day.
- A speedboat travels 100 miles in 2.5 hours.
- A helicopter goes 90 mph, but takes 3.5 hours to get to the harbor.
what should Ms. Croft do? Prepare a poster answering this question
with multiple representations of your thinking.
Answers
Answer:
The time it takes Ms. Croft to reach the cruise ship by speedboat = 6.11 hours
The time it takes Ms. Croft to reach the cruise ship by helicopter = 6.25 hours
since it takes less time for Ms. Croft to reach the cruise ship by taking the speedboat, her best choice to retrieve the artifact is to take the speedboat.
Step-by-step explanation:
The given parameters are;
The elapsed time by which Ms. Croft missed the boat = 5 hours
The speed of the cruise ship = 528 mi/day
The speed of the speedboat = 100 miles in 2.5 hours
The speed of the helicopter = 90 mph
The time it would take Ms. Croft to arrive at the harbor = 3.5 hours
Therefore, we have;
The cruise ship's speed = 528 miles per 24 hours = 528/24 mph = 22 mph
The location of the cruise after the first 5 hours = 5 × 22 = 110 miles
The speed of the speedboat = 100 miles per 2.5 hours = 100/2.5 = 40 mph
By traveling with a speed boat
The time at which Ms. Croft will intercept the cruise ship is given by the following relation;
40 mph × t = 22 mph × t + 110 miles
Which gives;
40 mph × t - 22 mph × t = 110 miles
18 mph = 110 miles
t = 110 miles/(18 mph) = 55/9 hours ≈ 6.11 hours
By taking the helicopter, we have;
Upon arrival at the harbor, after 3.5 hours, we find
90 mph × t = 22 mph × t + 22 mph × 3.5 hours + 110 miles
90 mph × t - 22 mph × t = 22 mph × 3.5 hours + 110 miles
68 mph × t = 77 miles + 110 miles = 187 miles
t = 187 miles/(68 mph) = 11/4 hours = 2.75 hours
The total time it takes Ms. Croft to reach the cruise ship by taking the helicopter = 3.5 + 2.75 = 6.25 hours
Therefore, since it takes less time for Ms. Croft to reach the cruise ship by taking the speedboat, her best choice to retrieve the artifact is to take the speedboat.
26 = z/19
20p question
Answers
Answer: I hope this helps
Step-by-step explanation:
26=z/19
Multiply both sides of the equation by 19
then you get
494=z but you need to
Swap the sides of the equation Which is something like this
z=494
Answer:
z = 494
Step-by-step explanation:
Given equation,
→ 26 = z/19
Now the value of z will be,
→ 26 = z/19
→ z/19 = 26
→ z = 26 × 19
→ [ z = 494 ]
Hence, the value of z is 494.
Graph the linear function w(x)=3/5x+2
Answers
Answer:
Answer: w= 3x+10 5x
Step-by-step explanation:
Let's solve for w. wx= 3 5 x+2 Step 1: Divide both sides by x. wx x = 3 5 x+2 x w= 3x+10 5x Answer: w= 3x+10 5x
Hope this helped :)
you purchase a set of 100 cube-shaped miniature magnets, each with an edge length of 1 } 8 inch. you arrange the cubes to form larger cubes, each one with a different edge length. how many cubes can you form? what is their total volume?
Answers
You can form a total of 38 different cubes using the set of 100 cube-shaped miniature magnets. The total volume of these cubes is 120.75 cubic inches.
To find the number of different cubes that can be formed, we need to consider the factors of 100. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. These factors represent the possible edge lengths of the cubes that can be formed.
Now, let's calculate the total volume of the cubes. Since each cube has an edge length of 1/8 inch, the volume of each cube can be calculated as (1/8)^3 = 1/512 cubic inches.
To find the total volume, we need to sum up the volumes of all the cubes. Let's calculate it:
Volume of cube with edge length 1/8 inch = (1/8)^3 = 1/512 cubic inches
Volume of cube with edge length 1/4 inch = (1/4)^3 = 1/64 cubic inches
Volume of cube with edge length 1/2 inch = (1/2)^3 = 1/8 cubic inches
Volume of cube with edge length 1 inch = (1)^3 = 1 cubic inch
We can calculate the total volume by adding up the volumes of these cubes:
Total volume = (1/512 + 1/64 + 1/8 + 1) * 38 = 120.75 cubic inches
Therefore, you can form a total of 38 different cubes using the set of 100 cube-shaped miniature magnets, with a total volume of 120.75 cubic inches.
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Problem
A store has 24 bicycles, which are evenly divided among 3 racks. How many bicycles are on each rack?
Answers
Eight..........................
Use the given information to prove the following theorem.
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
We let P be any point on line I, but different from point X.
Answers
The two right triangles formed by the perpendicular bisector of the segment \(\overline{YZ}\) are congruent, therefore, \(\overline{YP}\) is congruent to \(\overline{ZP}\) by CPCTC and YP = ZP by definition of congruence.
What are congruent triangles?Congruent triangles are triangles that have the same shape, sizes and interior angles.
The two column table to prove that a point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment can be presented as follows;
Statement \({}\) Reasons
\(\overline{PX}\) is the ⊥ bisector of \(\overline{YZ}\) Given
∠YXP = ∠ZXP = 90° \({}\) Definition of perpendicular bisector
ΔXYP and ΔXZP are right Δs\({}\) Definition
\(\overline{YX}\) ≅ \(\overline{ZX}\) \({}\) \({}\) Definition of bisected segment
\(\overline{XP}\) ≅ \(\overline{XP}\) \({}\) Reflexive property
ΔXYP ≅ ΔXZP \({}\) SAS congruence rule
\(\overline{YP}\) ≅ \(\overline{ZP}\) \({}\) CPCTC
YP = ZP \({}\) Definition of congruence
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The areas of similar triangles ΔABC and ΔDEF are equal. If the longest side of ΔDEF measures 25 units, what is the length of the longest side of ΔABC?
A.
5 units
B.
25 units
C.
75 units
D.
50 units
Answers
Answer:
It is 25 units
Step-by-step explanation:
The triangles are equal
Answer:
25 Units.
Step-by-step explanation:
The Triangles are equal and have the same area. Therefore, the longest side of ΔDEF and ΔABC will measure the same.
I need help ASAP IM IN A HURRY SO PLEASE
Answers
Step-by-step explanation:
Refer to pic...........
forty-four percent of us adults have little confidence in their cars. you randomly select twelve us adults. find the probability that the number of us adults who have little confidence in their cars is (1) exactly six and then find the probability that it is (2) more than 7.
Answers
The proportion of adults who have little faith in their automobiles, or the chance of success (p), is 44%.
How does probability work?The probability of an event occurring is referred to as probability. We frequently have to make predictions about the future in real life. We may or may not be aware of the outcome of an event.
At the point when this happens, we broadcast that there is plausible that the occasion will happen.
In conclusion, probability can be put to great use in business as well as in this rapidly expanding field of artificial intelligence. By simply dividing the total number of outcomes by the favorable number of possibilities, the probability of an event can be calculated using the probability formula.
Given that "p" stands for probability—that is, the fact that 45 percent of adults in the United States have little faith in their automobiles—p = 0.45, the likelihood of failure is q = 1-p = 1 - 0.45.
The chance of success (p), or the percentage of adults who have little faith in their automobiles, is therefore 44%, with q =0.55.
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i need help doing -12 plus -12 as a world wide explanation
Answers
Answer:
-24
Step-by-step explanation:
hope it helps have a good day/night :)
Pls give the correct answer
Answers
The secant of the angle is 765.
We are given that;
The point (7,−4) on the terminal side of the angle.
Now,
To find the secant of an angle, you need to know the length of the hypotenuse and the length of the side adjacent to the angle in a right triangle. The formula for the secant of an angle is:
secθ=hypotenuse/adjacent
where θ is the angle in standard position1.
To find the secant of the angle, you need to find the length of the hypotenuse and the length of the side adjacent to the angle. You can do this by using the Pythagorean theorem and the properties of a right triangle.
The hypotenuse is the distance from the origin to the point (7,−4). You can use the Pythagorean theorem to find it:
h^2=7^2+(−4)^2
h^2=49+16
h^2=65
The adjacent side is the horizontal distance from the origin to the point (7,−4). You can use the x-coordinate of the point to find it:
a=7
Now, you can substitute these values into the formula for secant and simplify:
secθ=hypotenuse/adjacent
secθ=765
This is already in simplest radical form, so you don’t need to do anything else.
Therefore, by trigonometric identities the answer will be 765.
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Chicken wire function drop down
Answers
is this a real question cause the awnser is fire
12. Sarah uses long division to convert
a rational number 2 to a decimal.
P
Which of the following statements
cannot be true?
A. The decimal form of P
q
terminates with a zero in the
tenths place.
B. The decimal form of P
eventually repeats.
C. The decimal form of never
terminates or repeats.
D. The decimal form ofis 0.
Answers
On solving thr provided question, by help of arithmetic progression, we got to know that a10 = 207
what is arithmetic progression?There are two approaches to define arithmetic progressions (AP): The difference between any two successive terms in an arithmetic sequence must always be equal, and it is sometimes referred to as a series in which all terms except the first are created by adding a predetermined number to the preceding term.
9,15,25..............
15-9 = 6
25-15 = 10
a4 - 25 = 14 => a4 = 39
frequently referred to as a series in which all terms save the first are formed by adding a predefined number to the preceding term since the difference between any two succeeding terms in the sequence must always be equal.
a5-39 = 18 => a5 = 57
a6-57 = 22 => a6 79
...
...
...
...
...
a10 - 169 = 38 => a10 = 207
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Helen is having a birthday
party. Her friends arrive at
her house at 8:15 p.m. They
leave three hours later. What
time do Helen's friends leave?
Answers
Answer: 11:15 p.m 5:15 p.m.
Step-by-step explanation:
The lengths of these sides of a triangle are 14cm, 22cm and 30cm. Find the measure of the smallest angle in the triangle to the nearest tenth.
Answers
As per given by the question,
There are given that a length of three sides of the triangle.
The length of sides are, 14cm, 22cm, and 30cm.
Now,
For finding the measure of the smallest angle,
The smallest angle in a triangle is always opposite the shortest side. and also the bigest angle is always opposite the longest side.
So,
Suppose length A is 14cm, B is 22 cm and C is 30cm.
Then,
From the law of cosines,
Let A be the smallest angle.
So,
\(14^2=30^2+22^2-2(30)(22)\cos A\)Now,
Find the value of angle A,
\(\begin{gathered} 14^2=30^2+22^2-2(30)(22)\cos A \\ 196=900+484-1320\cos A \end{gathered}\)Now,
\(\begin{gathered} 196=1384-1320\text{ cosA} \\ 196-1384+1320\cos A=0 \\ -1188+1320\cos A=0 \\ -1188=-1320\cos A \\ \cos A=\frac{1188}{1320} \end{gathered}\)Then,
\(\begin{gathered} \cos A=\frac{1188}{1320} \\ \cos A=0.9 \\ A=\cos ^{-1}(0.9) \\ A=25.84^{\circ} \end{gathered}\)Hence, the measure of the smallest angle is 25.84 degree.
Carl works at a veterinarian office. The first dog he sees is a Chihuahua who weighs 3 kilograms. The second dog he sees is a Great Dane who weighs 27 times as much as the Chihuahua. How much does the Great Dane weigh?
Answers
The Great Dane, being considerably larger than the Chihuahua, weighs a total of 81 kilograms.
Carl, who works at a veterinarian office, encounters two dogs. The first dog is a Chihuahua, and it weighs 3 kilograms.
The second dog, a Great Dane, is significantly larger than the Chihuahua.
To determine the weight of the Great Dane, we can use the given information that it weighs 27 times as much as the Chihuahua.
Since the Chihuahua weighs 3 kilograms, we can multiply this weight by 27 to find the weight of the Great Dane.
By performing this calculation (3 kilograms * 27), we discover that the Great Dane weighs 81 kilograms. Consequently, the Great Dane weighs 81 kilogrammes overall because it is much larger than the Chihuahua.
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x^2- 4x- 12 = 0
Solve by factoring and solve by completing the square
Answers
Use the formula (b/2)^2 in order to create a new term. Solve for by using this term to complete the square.
Cannot solve by completing the square.
There are two spinners containing only purple and orange slices.
Spinner A has 9 orange slices and 11 purple slices.
All the slices are the same size.
Spinner B has 8 orange slices and 8 purple slices.
All the slices are the same size.
Each spinner is spun.
List these events from least likely to most likely.
Event 1: Spinner B lands on a purple slice.
Event 2: Spinner B lands on an orange or purple slice.
Event 3: Spinner A lands on a white slice.
Event 4: Spinner A lands on an orange slice.
Least likely
Most likely
Event Event Event Event
Answers
Solution
Event 1: Probability is 8/16 = 1/2
Event 2: Probability is 1(certain to happen)
Event 3: Probability is 0(Impossible). There are no white slices in Spinner A
Event 4: Probability is 9/20
What condition has no solution?
Answers
A condition that is impossible to satisfy has no solution.
Mathematics can be a powerful tool for solving many problems, but there are some issues that can't be solved.
In particular, any condition that is impossible to satisfy, such as the logical statement ‘this statement is false’, has no solution in mathematics. This is because the statement can’t be proven true or false, and so it can’t be solved.
An unsolvable condition is one that cannot be solved using any combination of logical and mathematical steps.
This means that no matter how much time and effort is put in, a solution to the problem cannot be discovered.
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I need help with these
Answers
Answer:
87.7%
Step-by-step explanation:
Probability = number of positive results / total number of potential results
We will use the above formula to address the question of whether a random flight in Atlanta is likely to arrive on schedule.
Random flight in Atlanta is likely to arrive on time= the number of flights arriving in Atlanta on time, the cumulative number of flights arriving in Atlanta.
A measuring Task objective => To investigate your local athletics track to see wheather it is worked fairly for runners. who start on different lines.
Answers
The Task objective chosen is: A Measuring Task
The Objective is: To investigate the fairness of a local athlete's track for runners starting on different lines.
How do one carry about the investigation about?To carry about the investigation, follow the steps given below.
Choose a nearby sportsman's racetrack that you wish to examine.Split the pathway into several lanes, with a minimum of three lanes.Determine the length of each lane by utilizing either a measuring tape or measuring wheel. Make sure the measurements are precise.Place cones or markers to designate the starting points for each lane.Request the presence of a cohort of individuals who engage in running to take part in the study.Record the duration of each runner in completing the track for every lane.Evaluate the outcomes and contrast the timings of athletes beginning from distinct tracks.Closely examine the time variances among distinct lanes.Draw inferences from the examination. Inconsistent differences in lane times may indicate an unfair advantage for certain runners depending on their starting position.Lastly, make a crossword puzzle that includes mathematical vocabulary in a clever manner.
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```Class IX```
*Mathematics Project work*
*Do any one*
1. *A Measuring Task*
To investigate your local athletes track to see whether it is worked fairly for runners who start on different lines.
2. *Design a crossword puzzle with mathematical terms*
To review mathematics vocabulary, to give the opportunity for creative expression in designing puzzles, to act as a means of motivating the study of a given unit and to given recreation .
A coach buys a uniform and a basketball for each of the 20 players on the team. Each
basketball costs $10.40. The coach spends a total of $620 for uniforms and basketballs
What was the cost of a single uniform?
Answers
Answer:
$20.60
Step-by-step explanation:
You find how much it costs for the total amount of basketballs (10.40×20=208), then you subtract that number from the total cost of everything to get the cost of all the uniforms (620-208=412), then you divide the total costs for all uniforms by 20 to find the cost for a single uniform (412÷20=20.6). Always make sure that the decimals go until the hundredths place for USD.
Hope this helps!
a poll showed that 53.6% of americans say they believe that some psychics can help solve murder cases. what is the probability of randomly selecting someone who does not believe that some psychics can help solve murder cases.
Answers
The probability of randomly selecting someone who does not believe that some psychics can help solve murder cases is 46.4%. Hence, the answer is 46.4%.
Given the poll showed that 53.6% of Americans say they believe that some psychics can help solve murder cases.
We are to find the probability of randomly selecting someone who does not believe that some psychics can help solve murder cases.
Let us consider the following events:
Let A be the event that someone believes that some psychics can help solve murder cases and B be the event that someone does not believe that some psychics can help solve murder cases.
Then A and B are complementary events, that is A∩B=∅ (i.e., the events cannot happen at the same time) and A∪B = S (the event space).
Thus we can write P(A) = 53.6% and P(B) = 100% - 53.6% = 46.4%.
Therefore, the probability of randomly selecting someone who does not believe that some psychics can help solve murder cases is 46.4%. Hence, the answer is 46.4%.
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Solve the system of inequalities and indicate all the integers which are in the solution set: The answer isn't an inequality like -5
Answers
Answer:
{- 4, - 3, -2, -1, 0, 1, 2, 3}Step-by-step explanation:
Solve each inequality in the system:
3 - 2a < 13 ⇒ 2a > 3 - 13 ⇒ 2a > - 10 ⇒ a > - 55a < 17 ⇒ a < 17/5 ⇒ a < 3.4The solution is:
- 5 < a < 3.4The integer solution set is:
{- 4, - 3, -2, -1, 0, 1, 2, 3}\(\\ \rm\rightarrowtail 3-2a<13\)
\(\\ \rm\rightarrowtail -2a<10\)
\(\\ \rm\rightarrowtail -a<5\)
\(\\ \rm\rightarrowtail a>-5\)
And
\(\\ \rm\rightarrowtail 5a<17\)
\(\\ \rm\rightarrowtail a<17/5\)
\(\\ \rm\rightarrowtail a<3\dfrac{2}{5}\)
So
solution set
\(\\ \rm\rightarrowtail a\in\left(-5,3\dfrac{2}{5}\right)\)
So
a={-4,-3,-2,-1,0,1,2,3}