hilp meeee trying ....
Consider the two parabolas :y1=2x^2-3x-1 and y2=x^2+7x+20.
(a) Find the points of intersection of the parabolas and decide which one is greater than the other between the intersection points.
(b) Compute the area enclosed by the two parabolas.
(c) Use Mathcad to draw the graphs of the parabolas and find their intersection points graphically .
(d) Solve the differential equation dy/dx = -exp(5-x)+3/5x^2 ; y(5)=3
To solve the given problems, we will break them down step by step. We will start with problem (a).
(a) Find the points of intersection of the parabolas and decide which one is greater than the other between the intersection points.
To find the points of intersection of the parabolas, we need to set the two equations equal to each other and solve for x.
2x^2 - 3x - 1 = x^2 + 7x + 20
Rearranging the equation:
x^2 - 10x - 21 = 0
Now we can use the quadratic formula to find the values of x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a = 1, b = -10, and c = -21. Plugging these values into the formula, we get:
x = (10 ± √(10^2 - 4(1)(-21))) / 2(1)
Simplifying further:
x = (10 ± √(100 + 84)) / 2
x = (10 ± √184) / 2
This gives us two possible values for x. To find the corresponding y-values, we can substitute each value of x back into one of the original equations.
For x = (10 + √184) / 2:
y = 2((10 + √184) / 2)^2 - 3((10 + √184) / 2) - 1
Simplifying this expression will give us the y-value for the first intersection point.
Similarly, for x = (10 - √184) / 2:
y = 2((10 - √184) / 2)^2 - 3((10 - √184) / 2) - 1
Simplifying this expression will give us the y-value for the second intersection point.
To determine which parabola is greater between the intersection points, we compare the y-values of the two points. The parabola with the higher y-value at the intersection point is considered greater.
(b) Compute the area enclosed by the two parabolas.
To compute the area enclosed by the two parabolas, we need to find the x-values where the parabolas intersect and integrate the difference in their y-values over that interval. The absolute difference between the two y-values represents the height of the enclosed region at each point.
Integrate the absolute difference of the two functions over the interval of intersection to find the area.
(c) Use Mathcad to draw the graphs of the parabolas and find their intersection points graphically.
To draw the graphs of the parabolas and find their intersection points graphically, we can use a graphing software or program like Mathcad. Input the equations y1 = 2x^2 - 3x - 1 and y2 = x^2 + 7x + 20 into the graphing tool, and it will generate the graphs of the two parabolas. The intersection points can be identified visually by observing where the two graphs intersect.
(d) Solve the differential equation dy/dx = -exp(5-x) + 3/5x^2; y(5) = 3.
To solve the given differential equation, we need to separate the variables and integrate:
dy / (-e^(5-x) + 3/5x^2) = dx
Now we integrate both sides of the equation:
∫(1 / (-e^(5-x) + 3/5x^2)) dy = ∫dx
Integrating the left side of the equation is a bit complicated due to the presence of exponential and polynomial terms. We need to find the antiderivative of the expression, which might require advanced techniques like partial fractions or numerical methods.
After integrating the left side, we integrate the right side as well.
Once we have the general solution, we can use the initial condition y(5) = 3 to find the value of the constant of integration and obtain the particular solution for the given initial condition.
Solving this differential equation might require the use of a symbolic math software like Mathcad or numerical methods like Euler's method or Runge-Kutta methods if analytical solutions are not feasible.