Answer:
Part I)
[tex]a=-1\text{ and } c=2[/tex]
Giving the equation:
[tex]y=-x^3+x+2[/tex]
Part II)
(-1, 2)
Step-by-step explanation:
We have the equation:
[tex]y=ax^3+x+c[/tex]
Where a and c are constants.
We are given gradient/slope of the curve at the point (1, 2) is -2.
Part I)
We know that the gradient of the curve exactly at the point (1, 2) is -2.
So, we will differentiate our equation. We have:
[tex]\displaystyle y=ax^3+x+c[/tex]
Differentiate both sides with respect to x:
[tex]\displaystyle y^\prime=\frac{d}{dx}\big[ax^3+x+c\big][/tex]
Expand the right:
[tex]\displaystyle y^\prime=\frac{d}{dx}\big[ax^3]+\frac{d}{dx}\big[x\big]+\frac{d}{dx}\big[c\big][/tex]
Since a and c are simply constants, we can move them outside:
[tex]\displaystyle y^\prime=a\frac{d}{dx}\big[x^3]+\frac{d}{dx}\big[x\big]+c\frac{d}{dx}\big[1\big][/tex]
Differentiate. Note that the derivative of a constant is simply 0.
[tex]\displaystyle y^\prime=3ax^2+1[/tex]
We know that at the point (1, 2), the gradient/slope is -2.
Hence, when x is 1 and when y is 2, y’ is -2.
So (we don’t have a y so we can ignore it):
[tex]-2=3a(1)^2+1[/tex]
Solve for a. Simplify:
[tex]-2=3a+1[/tex]
Subtract 1 from both sides yield:
[tex]-3=3a[/tex]
So, it follows that:
[tex]a=-1[/tex]
Therefore, our derivative will be:
[tex]y=-3x^2+1[/tex]
Now, we can go back to our original equation:
[tex]y=ax^3+x+c[/tex]
Since we know that a is -1:
[tex]y=-x^3+x+c[/tex]
Recall that we know that a point on our curve is (1, 2).
So, when x is 1, y is 2. By substitution:
[tex]2=-(1)^3+1+c[/tex]
Solve for c. Simplify:
[tex]2=-1+1+c[/tex]
Then it follows that:
[tex]c=2[/tex]
Hence, our entire equation is:
[tex]y=-x^3+x+2[/tex]
Part II)
Our derivative is:
[tex]y^\prime=-3x^2+1[/tex]
If the gradient is -2, y’ is -2. Hence:
[tex]-2=-3x^2+1[/tex]
Solve for x. Subtracting 1 from both sides yields:
[tex]-3=-3x^2[/tex]
So:
[tex]x^2=1[/tex]
Then it follows that:
[tex]x=\pm1\\[/tex]
Since we already have the point (1, 2) where x is 1, our other point is where x is -1.
Using our equation:
[tex]y=-x^3+x+2[/tex]
We can see that our second point is:
[tex]y=-(-1)^3+(-1)+2[/tex]
Simplify:
[tex]y=1-1+2=2[/tex]
So, our second point where the gradient is -2 is at (-1, 2).
I’m very confused on this! Where do I even start on here?? This is extra credit for my geometry class !
A car has 15 gallons of gas in its tank. The car travels 35 miles per gallon of gas. It uses 1/35 of a gallon of gas to go 1 mile.
How far can the car travel with 15 gallons? Show your reasoning.
How much gas does the car use to go 100 miles? Show your reasoning.
HELP ME PLEASE THIS QUESTION IS WORTH 65% OF MY GRADE!!!!
Answer:
20 gallons hope this helped
x²=9/25
what is it
need help reply
a plane flies 480 miles with a 15 mph tailwind and then back 480 miles with a 15 mph headwind. if the plane completes the trip in 7.2 hours, how fast is the plane traveling against the wind and with the wind
simplyfy: 4x + 8 + x -2 -3x
Answer:
2x+6
Step-by-step explanation:
To simplify an expression such as this you must add like terms. Like terms are terms that have the same variable and power. In this expression 4x, x, -3x are like terms and -2 and 8 are also like terms. To add like terms simply add the coefficients and remember adding a negative is the same as subtracting. So, add the previously listed terms 4x + x + (-3x) = 2x, then -2+8= 6. This means the simplified expression is 2x+6.
There are 12 tomato plants in a garden. Each plant has 7 branches ad each branch has 4 tomatoes growing on it. If one-third of the tomatoes are picked, how many tomatoes were picked
Answer:
112
Step-by-step explanation:
4*7=2828*12=336336*(1/3)=112If one-third of the tomatoes are picked, the number of tomatoes that were picked is 112 tomatoes.
Number of picked tomatoes:Number of picked tomatoes=(Growing tomatoes× Tomatoes branch× Tomatoes plant)× One-third
Number of picked tomatoes=(4×7×12)×1/3
Number of picked tomatoes=336×1/3
Number of picked tomatoes=112 tomatoes
Inconclusion if one-third of the tomatoes are picked, the number of tomatoes that were picked is 112 tomatoes.
Learn more about plants here:https://brainly.com/question/600212
2(6.75)+1.03+3S=22 therefore 3S = 14.53 divide by 3 so S equals 4.84 per pair of socks
Answer:
each pair of socks cost $2.49
Step-by-step explanation:
Eric spent $22.00, including sales tax on 2 jerseys and 3 pairs of socks. The jerseys cost $6.75 each and the total sales tax was $1.03. How much did each pair of socks cost?
Solution:
The total money spent was $22.00. 2 jerseys costing $6.75 each was bought, the total tax was $1.03. 3 pairs of socks was bought, Let S represent the cost for each pair of socks. The total money spent is the sum of the money spent on jersey, socks and tax. Hence:
2(6.75) + 3S + 1.03 = 22
13.5 + 3S + 1.03 = 22
14.53 + 3S = 22
3S = 22 - 14.53
3S = 7.47
S = 2.49
Hence each pair of socks cost $2.49
Write an equation for the translation of y=|x|
Answer:
y=1x
Step-by-step explanation:
Which of the following equations is equivalent to 4x+2y=12
A. 2x+2y=6
B. 8x+2y=24
C. 2x+y=6
D. 4x+4y=24
Answer:
my answer lies in paper
2. MULTIPLE CHOICE What is the y-intercept when 3x – 2y = -6 is graphed?
Answer:
D. 3
Step-by-step explanation:
Subtract 3x from both sides. Divide both sides by -2. Simplify. You get
y = 3/2x + 3
3 is the y-intercept.
~theLocoCoco
At a basketball camp there are 4 girls for every 5 boys. What is the ratio for BOYS to TOTAL number of campers?
There are nine campers in total. There are five boys in the camp. Here are the details:
Ratio: 5:9
Fraction: 5/9
Decimal: 0.5555...........
Percentage: 56%
Hope this helped! Have a wonderful Day! :D
What is the value of the expression when a = –5 and c = –1? The expression is 8(a + 2c)
Answer:
8(a + 2c) = -56
Step-by-step explanation:
Given that,
An expression : 8(a + 2c)
We need to find the value of the above expression when a = -5 and c = -1.
Put the values of a and c in the given expression.
8(a + 2c) = 8(-5+2(-1))
= 8(-5-2)
= 8(-7)
= -56
Hence, the value of the given expression is -56.
Translate the inequality and solve. Three times a number is greater than six. 1 point
Answer:
3x > 6
x > 2
Step-by-step explanation:
A store sells 6 T-shirts for $59.34. What is the unit cost per T-shirt?
OA. $9.89
OB. $8.48
OC. $53,34
OD $0.10
Answer:
$9.89
Step-by-step explanation:
Find the 10th term of the geometric sequence show below
3x^7,9x^9, 27x^11...
PLEASE ANSWER!! TY
Answer:
t_10 = 59049 x^25
Step-by-step explanation:
a = 3x^7
r = 3 x^2
t_10 = (3x^7)(3x^2)^10
t_10 = 3x^7 3^(10-1) * (x^2)^(10 - 1)
t_10 = 3x^7 * 3^9 * (x^2)^9
t_10 = 3x^7 * 19683 x^18
t_10 = 59049 x^25
find the coordinates of the point which divides the line segment joining the points (4, -3) and (8.5) in the ratio 3:1 internally
Answer:
The coordinates of the division point are (7, 3)
Step-by-step explanation:
The coordinates of the point (x, y) which divides a segment with endpoints (x1, y1) and (x2, y2) at a ratio m1: m2 internally is ([tex]\frac{m1x2+m2x1}{m1+m2}[/tex] , [tex]\frac{m1y2+m2y1}{m1+m2}[/tex] ).
∵ The line segment joining the points (4, -3) and (8, 5)
∴ x1 = 4 and x2 = 8
∴ y1 = -3 and y2 = 5
∵ The point divides the segment at ratio 3: 1
∴ m1 = 3 and m2 = 1
→ By using the rule above
∵ x = [tex]\frac{(3)(8)+(1)(4)}{3+1}[/tex] = [tex]\frac{24+4}{4}[/tex] = [tex]\frac{28}{4}[/tex]
∴ x = 7
∵ y = [tex]\frac{(3)(5)+(1)(-3)}{3+1}[/tex] = [tex]\frac{15-3}{4}[/tex] = [tex]\frac{12}{4}[/tex]
∴ y = 3
∴ The coordinates of the division point are (7, 3)
Which line shows the graph of y = 4x - 5 reflected over the y-axis?
y = 4x + 5
y = 4x - 5
Y=2x-5
y = -2x + 5
Answer:
The line y = -4x - 5 shows the reflection over the y-axis.
From the attached graph, the blue line i.e. y = -4x-5 represents the reflection over the y-axis.
Step-by-step explanation:
When we reflect a line function over the y-axis, all we need is to negate the x-coordinate or value of 'x', while y-coordinate or y-value remains the same.
Given the function
y = 4x-5
replace x with -x
y = 4(-x) - 5
y = -4x - 5
Thus, the line y = -4x - 5 shows the reflection over the y-axis.
From the attached graph, the blue line i.e. y = -4x-5 represents the reflection over the y-axis.
From the graph, it is clear that that that x-coordinate of y = 4x-5 is negated.
como simplifico las fracciones de 14+8
Answer:
Simplificadora de Fracciones
Si dividirmos 14 entre 8 obteremos un entero 1 más una fracción propria 6/8. Así, la fracción equivalente es el NÚMERO MIXTO 1 3/4). La fracción 14/8 es igual a 14÷8 y también se puede expresar por el número decimal 1.75.
Step-by-step explanation:
Express 25 as power with base 5
Answer: express a number in base 5 notation you need to write it in powers of 5. ... 52 = 25 53 = 125 54 = 625. 625 is larger than 613 so I only need a 53 digit, a 52
Step-by-step explanation:
Answer:
POWERS OF 5:
.52 = 25
53 = 125
54 = 625
So 625 is larger than 613
so you will only need 53
but then you also will have 52 so you have 52 as your answer i hope this helps :)
Evan’s family drove to a theme park for vacation.
They drove the same speed throughout the trip.
The first day, they drove 300 miles in 6 hours.
The second day, they drove 250 miles in 5 hours.
The third day, they arrived at the park after driving for 3 hours.
How many miles did they drive on the third day?
- First: Calculate how many miles the family is driving in ONE hour.
There are two equations you can form to solve this.
300 / 6 = mph (miles per hour)
250 / 5 = mph (miles per hour)
300 / 6 = 50
250 / 5 = 50
They are driving 50mph. Now, you need to find out how many miles they drive in three hours.
50 * 3 = 150
Evan's family drove 150 miles on the third day.
Answer:
150 miles
Step-by-step explanation:
The first day, they drove 300 miles in 6 hours so divide 6 by 300. 300 divided by 6 equals fifty. On the third day, they drove half the amount of time, so it would be half the amount of miles because they drove the same speed throughout the trip.
300/2 so the answer to the question is 150 miles.
i need help with pre-algebra, how do you convert exponential expressions? please help!
Answer:
Go on You Tube and look up
Converting Exponential Equations to Logarithmic Equations - College Algebra Tips
...................helppppp
A store is having a sale with 10% off everything.
b) Does your equation for part a represent a proportional relationship?
Answer:
what equation can you give more through instructions please
please help... my teacher makes me do this thing called mathspace and im due 5 tasks can you help me with this question.
A bag of manure weighs $110kg. If the weight of the bag increases by $30% find the new weight of the bag.
Answer:
143 kg
I'm not sure but I think this is correct
Step-by-step explanation:
Using the greatest common factor for the terms, how can you write 56+32 as a product?
Step-by-step explanation:
56 + 32
= 8 * 7 + 8 * 4
= 8(7 + 4)
= 8 * 11
= 88.
Hi, i need help please.
Answer:
The answer is A.8x+13
Step-by-step explanation:
Help need in 29 minutes
Answer:
d)
c)
Step-by-step explanation:
any more help just ask :)
6
[tex] \frac{?}{?} [/tex]
m+4=1½
Answer:
i don't get the question ...
Step-by-step explanation:
Can someone help me?
Answer:
The probability that the pet seen was sick is 53.57%.
Step-by-step explanation:
More than 200 universities in the United States and many more worldwide have sport management programs. Approximately how many are accredited by the Commission on Sport Management Accreditation (COSMA)
Answer:
54 programs
Step-by-step explanation:
As of today's date 12/25/2020, there are a total of 54 programs that have been accredited by the Commission on Sport Management Accreditation (COSMA). Each university program that has been accredited can be found on COSMA's official website along with the date in which each program was officially accredited and the basic information regarding the program and its university. More and more programs are being added as time goes by.