Consider the differential equation dy/dx = 1/3x(y-2)^2.
(a) A slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point (0, 2), and sketch the solution curve that passes through the point (1, 0).
(b) Let y = f(x) be the particular solution to the given differential equation with initial condition f(1) = 0. Write an equation for the line tangent to the graph of y = f(x) at x = 1. Use your equation to approximate f(0.7).
(c) Find the particular solution y = f(x) to the given differential equation with initial condition f(1) = 0.
(This
was on the no-calculator section of the recently-released AP Calculus
AB 2018 exam so I appreciate it if you tried to limit calculator usage)
To sketch the solution curves passing through (0, 2) and (1, 0), we can use the slope field provided. The tangent line to y = f(x) at x = 1 can be found by taking the derivative of f(x) and substituting x = 1. We can find the particular solution to the differential equation by integrating it with the given initial condition f(1) = 0.
Explanation:(a) Solution curves through (0, 2) and (1, 0)To sketch the solution curves, we can use the slope field provided. For the point (0, 2), the slope is 1/3(0)(2-2)^2 = 0. Thus, at (0, 2) the solution curve is horizontal. For the point (1, 0), the slope is 1/3(1)(0-2)^2 = 8/3. Thus, at (1, 0) the solution curve is directed upwards with a slope of 8/3.
(b) Tangent line at (1, 0) and f(0.7)Since f(1) = 0, we already know that the point (1, 0) lies on the graph of y = f(x). The equation of the tangent line to y = f(x) at x = 1 can be found by finding the derivative of f(x) and substituting x = 1. In this case, the derivative is dy/dx = 1/3x(y-2)^2, so at x = 1, we have dy/dx = 1/3(1)(0-2)^2 = -8/3. Therefore, the equation of the tangent line is y - 0 = (-8/3)(x - 1), which simplifies to y = -8/3(x-1). To approximate f(0.7), we can substitute x = 0.7 into the equation of the tangent line to get y = -8/3(0.7 - 1) = -8/3(0.3) = -8/10 = -0.8.
(c) Find the particular solution with f(1) = 0To find the particular solution, we can integrate the differential equation with the given initial condition. Starting with dy/dx = 1/3x(y-2)^2, we can rewrite it as (y-2)^(-2)dy = 1/3xdx. Integrating both sides will give us the equation of the particular solution.
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Below is a graph that represents the total profits for a third month. Write the equation of the line that represents this graph. Show your work or explain how you determined the equations.
PLEASE HELP ASAP!! NEED NOW! THANKS!
y=-4/3x+300 i hope this helps you
Mike is looking for a loan. He is willing to pay no more than an effective rate of 8.000% annually. Which, if any, of the following loans meet Mike’s criteria? Loan X: 7.815% nominal rate, compounded semiannually Loan Y: 7.724% nominal rate, compounded monthly Loan Z: 7.698% nominal rate, compounded weekly a. Y only b. X and Z c. Y and Z d. None of these meet Mike’s criteria.
Answer:
b. X and Z
Step-by-step explanation:
Since, the effective rate is,
[tex]r=(1+\frac{i}{n})^i-1[/tex]
Where, i is the nominal rate,
n is the number of compounding periods,
For loan X,
i = 7.815 % = 0.07815,
n = 2, ( 1 year = 2 semiannual )
Thus, the effective rate would be,
[tex]r=(1+\frac{0.07815}{2})^2-1[/tex]
[tex]=0.079676855625[/tex]
[tex]=7.9676855625\%\approx 7.968\%[/tex]
Since, 7.968 % < 8.000 %,
⇒ Loan X meets Mike's criteria,
For loan Y,
i = 7.724 % = 0.07724,
n = 12 ( 1 year = 12 months ),
Thus, the effective rate would be,
[tex]r = ( 1+\frac{0.07724}{12})^{12}-1[/tex]
[tex]=0.0800339518197[/tex]
[tex]=8.00339518197\% \approx 8.003\%[/tex]
Since, 8.003 % > 8.000 %,
⇒ Loan Y does not meet his criteria,
For loan Z,
i = 7.698 % = 0.07698,
n = 52 ( 1 year = 52 weeks ),
Thus, the effective rate would be,
[tex]r = (1+\frac{0.07698}{52})^{52}-1[/tex]
[tex]=0.0799589986135[/tex]
[tex]=7.99589986135\%[/tex]
[tex]\approx 7.996\%[/tex]
Since, 7.996 % < 8.000 %,
⇒ Loan Z meets his criteria.
Therefore, Option 'b' is correct.
If one can of paint contains 2 2/5 gallons, how much paint is in 12 cans of paint?
Answer:12 2/4
Step-by-step explanation:
First of all that eqation is improper. So lets make that proper. To do that you just count by five till you get close to the number. 5,10,15,20 im close so lets stay there. So then it would be 4 1/5. Then you have to multiple that by 12.
And your awnser would be 50/4
then you need to make it propper.
12 2/4
27.3 less than k is -57.7. What is the value of k ?
which set of number could represent the length of the sides of a right triangle? A 9 11 14 B 15 18 21 C 7 24 25 D 8 9 10
A store purchases an item wholesale for 60$ and marks it up by 70 percent if there is a 7 percent tax on the item how much will the item cost
Answer:
109.14
Step-by-step explanation:
to Start off do 60 x .70, then add the answer to 60, then multiply by .07 then add that to 102, for the answer of 109.14
Q # 13 please Graph the image. of AMOP for the translation
which has the smallest value 11 , 0.675,1.23,0.42,1.011
A bag contains 7 blue, 12 red, and 6 orange marbles. Find each probability if you draw one marble at random from the bag. Write as a fraction. a) P (purple)
b) P (red or orange) Please
c)P(not blue) Help
priority question
The dot plots below show the test scores of sixth- and seventh-grade students: Dot plot for Grade 6 shows 6 dots on score 70, 4 dots on score 80, 6 dots on score 90, and 4 dots on score 100. Dot plot for Grade 7 shows 5 dots on score 50, 7 dots on score 60, 4 dots on score 70, and 4 dots on score 80. Based on visual inspection of the dot plots, which grade, if any, appears to have the higher mean score? Grade 6 Grade 7 Both groups show about the same mean score. No conclusion about mean score can be made from the data.
Answer:
Grade 6 i think
What is the y-intercept of the equation of the line that is perpendicular to the line y = 3/5x + 10 and passes through the point (15, –5)?
the radius of a glass is 1.25 inches. What circumference? use 3.14 for pi
select true statements. (-10x^(2)+5x+3)/(2x^(2)+4)
Answers are
The numerator has 3 terms
The denominator includes a coefficient of 4
The denominator includes a coefficient of 2
The numerator has 2 terms
The numerator includes a coefficient of 3
Answer: True statements:: The numerator has 3 terms; The denominator includes a coefficient of 2.
Step-by-step explanation:
What is the sum of the roots of the polynomial shown below f(x)=x^3-4x^2-11x+30
you have a part-time job that pays $4.75 an hour with tips allergy $4 and fifty an hour your deductions are fic a 7.65% federal tax withholding 12.5% in state tax withholding 6.85 % you work 20 hours per week you want to save $40 a week how much left in your discretionary income per week soon that all the income from your part time job is discretionary and the next one is
You decide to save 15% of your realized income how much do you save per month?
What is the slant height x of this square pyramid? The figure shows a square pyramid. The slant height is shown as a dashed line perpendicular to the base edge and is labeled as x. The length of the lateral edge is 4 meters. The lateral edge makes a 60 degree angle with the base edge. Enter your answer in the box. Express your answer in radical form.
Based on the information given, it should be noted that the slant height will be 2✓3 meters.
From the information given, the slant height is shown as a dashed line perpendicular to the base edge and is labeled as x as the length of the lateral edge is 4 meters and the lateral edge makes a 60 degree angle with the base edge.Therefore, the calculation of the slant height goes thus:
Tan 60° = x/2
x = 2 × tan 60°
x = 2 × ✓3
x = 2✓3
Learn more about triangles on:
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find the value of x
Answer:
43 degrees
Step-by-step explanation:
You can see that the triangle is congruent, so x is equal to 43. dont forget to include degrees!
average of 2.99 2.59 3.50
Answer:
6.74
Step-by-step explanation:
add all numbers and divide how many numbers there are.
summation to integral? calculus
Final answer:
In calculus, summation involves adding discrete values, while integration is akin to adding an infinite number of infinitesimally small products to find areas, volumes, etc. Integral calculus extends these concepts to continuous functions and is used, for example, to find position from acceleration by integrating twice.
Explanation:
The transition from summation to integral in calculus is an important concept that bridges the discrete world with the continuous. When we summate, we're adding together values to find a total amount. In a similar vein, an integral can be thought of as a continuous sum, where we're adding up infinitely small products to calculate an area, volume, or other quantities.
When performing an integral, the dimensions of what we're integrating must be consistent. For example, if we're integrating velocity (v) with respect to time (t), we obtain a dimension that is the product of the two, such as distance. This principle is rooted in the requirement of dimensional consistency, analogously understood in the phrase "You can't add apples and oranges."
Understanding the relationship between discrete sums and continuous integrals is crucial for applications in various fields, such as physics where integrals can represent cumulative effects over time or space. For example, integrating an acceleration function gives us velocity, and further integration yields position. This process allows us to work backward from acceleration to position, utilizing the power of integral calculus.
Using known taylor series, find the first four nonzero terms of the taylor series about 0 for the function. sqrt((1 + t))sin(t)
Answer:
T series
Step-by-step explanation:
use similar triangles to solve. a person who is 5 feet tall is standing 140 ft from the base of a tree and the tree cast a 154 foot shadow. the person's Shadow is 14 ft in length what is the height of the tree
Q # 18,Graph the inequality on a coordinate plane, - y < 3 x - 5
Hey can you please help me posted picture of question
The home of Russell Slater is assessed at $140,000. The tax rate is 11.8 mills. The actual amount of tax on Russell's home is:
$1,652
$1,462
$1,362
None of these
$1,562
if 7 out of 8 people use crust toothpaste, how many use this product in a city with a population of 40,000?
Solve each system of equation algebraically
y=x+2
y=-3x
The easiest method to use to solve this equation is substitution. This is because we already have our second equation set equal to the variable y, so we can substitute the x values in for the variable y in the first equation, as follows:
y = x + 2
y = -3x
-3x = x + 2
To simplify, we should subtract x from both sides of the equation so that we can have all of the x terms isolated on the left side of the equation.
-3x - x = x - x + 2
-4x = 2
Finally, we must divide both sides of the equation by -4 to get the variable x alone.
x = -2/4
Because both the numerator and denominator of this fraction are divisible by 2, we can use this knowledge to simplify the fraction by dividing by the GCF of 2.
x = -1/2
Now that we know the value of x, we can substitute this value into either of our original equations to find out the value of y.
y = -3x
y = -3(-1/2)
y = 3/2
Therefore, your answer is x = -1/2 and y = 3/2, or written as an ordered pair (-1/2, 3/2).
Hope this helps!
Esther pays $467 per month for 5 years for a car. she made a down payment of $3,700. if the loan costs 7.1% per year compounded monthly, what was the cash price of the car?
The correct cash price is d) $27,228.33.
To determine the cash price of the car, we need to calculate the present value of all payments made by Esther, including her down payment.
Calculate the total number of monthly payments:
Total number of payments: 5 * 12 = 60
Determine the monthly interest rate:
Monthly interest rate: 0.071 / 12 = 0.0059167 (approximately)
Calculate the present value of the monthly payments using the formula for the present value of an annuity:
P = PMT × [tex]\frac{[1 - (1 + r)^{-n]} }{ r}[/tex]
P = 467 × [tex]\frac{ [1 - (1 + 0.0059167)^{-60}]}{ 0.0059167}[/tex] = $23,528.33
Add the down payment to the present value of the monthly payments to get the total cash price of the car:
Total cash price = $23,528.33 + $3,700 = $27,228.33
Therefore, the correct answer is (d) $27,228.33.
The complete question is
Esther pays $467 per month for 5 years for a car. she made a down payment of $3,700. if the loan costs 7.1% per year compounded monthly, what was the cash price of the car?
a) $23,528.33
b) $19,828.33
c) $29,820.57
d) $27,228.33
e) $37,220.57
Q # 4 find the slope of the line through the pair of points A(2, -3), P(2, 9)
Eve is replacing the soil in her plant containers. How many cubic inches of soil will Eve need for the two rectangular containers?
A.45
B.216
C.648
D.864