1. The general solution of the differential equation dy - 0.2x dx = 0 is a family of curves. These curves are all:
a. lines
b. hyperbolas
c. parabolas (my answer)
d. ellipses
2. The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of 2∫1 f(x) dx .
x 1 1.1 1.3 1.6 1.7 1.8 2.0
f(x) 1 3 5 8 10 11 14
no, y' = 0.2x
so, yes, a parabola.
dy/dx = 0.2
constant slope is a parabola ???
first trap = (1/2)(1+3)(1.1 - 1)
second = (1/2)(3+5)(1.3 -1.1)
etc
whoops ! Go with Steve
1. To find the general solution of the differential equation dy - 0.2x dx = 0, we can use separation of variables.
First, we can rearrange the equation to isolate the variables:
dy = 0.2x dx
Then, we integrate both sides with respect to their respective variables:
∫ dy = ∫ 0.2x dx
The integral of dy is simply y, and the integral of 0.2x dx can be found by applying the power rule of integration:
y = 0.2 * (1/2)x^2 + C
where C is the constant of integration.
The general solution of the differential equation is a family of curves represented by the equation y = 0.1x^2 + C, where C can take on any constant value.
Now, looking at the answer choices provided, we can see that the correct answer is c. parabolas.
2. To approximate the value of 2∫1 f(x) dx using a trapezoidal estimation with 6 trapezoids, we can follow the steps below:
1. Divide the interval [1, 2] into 6 equal subintervals. The width of each subinterval can be calculated as (2 - 1) / 6 = 1/6.
2. Evaluate the function f(x) at the endpoints and the midpoints of each subinterval. Record the values in the table below:
x | f(x)
------------------
1.0 | 1
1.1 | 3
1.2 | 5
1.3 | 8
1.4 | 10
1.5 | 11
1.6 | 14
1.7 | 10
1.8 | 10
1.9 | 11
2.0 | 14
3. Calculate the sum of the areas of the trapezoids using the formula:
Area of trapezoid = (1/2) * (y1 + y2) * Δx
where y1 and y2 are the function values at the endpoints of each subinterval, and Δx is the width of each subinterval.
Sum up the areas of all 6 trapezoids:
Area = (1/2) * (1 + 3) * (1/6) + (1/2) * (3 + 5) * (1/6) + (1/2) * (5 + 8) * (1/6) + (1/2) * (8 + 10) * (1/6) + (1/2) * (10 + 11) * (1/6) + (1/2) * (11 + 14) * (1/6)
Simplify the calculation:
Area = 20/6 = 10/3
4. Finally, multiply the result by 2 to get the approximation of 2∫1 f(x) dx:
2 * (10/3) = 20/3 approximately equal to 6.67
Therefore, the value of 2∫1 f(x) dx using a trapezoidal estimation with 6 trapezoids is approximately 6.67.