Let f be the function that contains the point (-1,8) and satisfies the differential equation dy/dx=10/(x^2+1)
(a) Write the equation of the tangent to f at x=-1.
(b) Use your equation in part a to estimate f(0).
(c) We know that the integral from -1 to 0 of 10/(x^2+1) equals about 7.854 and the integral from a to b of f(x)dx = F(b)-F(a). Use this to find f(0).
I think i did parts a and b right. i got y-8=5(x+1) for part a and f(0)=13 for part b, but i don't know how to do part c at all.
Thank you!
(a)
well, we have a point and a slope (5), so
y-8 = 5(x+1)
(b)
dy = 10/(x^2+1) dx, so if dx=1,
dy = 10/2 (1) = 5
So, the estimate is 8+5 = 13
(c)
F(0)-F(-1) = 7.854
F(0) = 7.854 + F(-1) = 7.854+8 = 15.854
I hope that (c) had a typo and didn't want f(0), since f(0) = 10. (f(x) = dy/dx)
it wasn't a typo.. how did you get 10?
well, they told us that f(x) = 10/(x^2+1)
Looks to me like f(0) = 10/(0+1) = 10
(a) Well, your equation y - 8 = 5(x + 1) is almost correct! Just do a little Clown Bot adjustment. Let me tell you, your equation has the right idea but some numbers gone wild. So let's put it back into shape. The equation of the tangent to f at x = -1 is y - 8 = 10(x + 1).
(b) Now, let's estimate f(0) using the equation we found in part a. Plug in x = 0 and solve for y. We have y - 8 = 10(0 + 1), which simplifies to y - 8 = 10. Add 8 to both sides and we get y = 18. So f(0) is estimated to be 18. Good job, you're getting the hang of this!
(c) Now for part c, we'll need to use a little brainpower here. We know that the integral from -1 to 0 of 10/(x^2+1) equals about 7.854, but we want to find f(0). Here's my clowntastic plan: let's integrate dy/dx = 10/(x^2+1) to find an expression for f(x). Once we have f(x), we can evaluate it at x = 0 to find f(0). Are you ready? Here we go!
First, let's integrate dy/dx = 10/(x^2+1) with respect to x. We get ∫dy = ∫10/(x^2+1) dx. Integrating both sides, y = 10∫1/(x^2+1) dx. Using a little clown magic, the integral of 1/(x^2+1) is arctan(x). So we have y = 10arctan(x) + C, where C is a constant we need to determine.
To find C, we can use the point (-1, 8) given in the problem. Plugging in x = -1 and y = 8 into our expression for y, we have 8 = 10arctan(-1) + C. The arctan(-1) is -π/4 (approximately -0.7854), so we can rewrite the equation as 8 = -7.854 + C. Solving for C, we find C = 15.854. Now we have the expression for f(x)!
Plugging in x = 0, we get f(0) = 10arctan(0) + 15.854. The arctan(0) is 0, so our final estimation for f(0) is 15.854. Bravo, you did it!
To find the equation of the tangent to f at x = -1, we need to use the slope-intercept form of a line.
(a) The differential equation provided gives us the slope of the tangent line at any point (x, y) on the curve f. So, to find the slope of the tangent line at x = -1, we substitute x = -1 into the given differential equation:
dy/dx = 10/(x^2+1)
dy/dx = 10/((-1)^2+1)
dy/dx = 10/2
dy/dx = 5
Therefore, the slope of the tangent line at x = -1 is 5. Now we can use the point-slope form of a line to write the equation of the tangent line. Given the point (-1, 8) and the slope m = 5, the equation of the tangent line is:
y - 8 = 5(x + 1)
So, you correctly found the equation of the tangent line for part (a).
Now, let's move on to part (b).
(b) To estimate f(0) using the equation of the tangent line at x = -1, we substitute x = 0 into the equation of the tangent line:
y - 8 = 5(0 + 1)
y - 8 = 5
y = 5 + 8
y = 13
So, you correctly calculated f(0) as 13 for part (b).
Finally, for part (c):
(c) We are given that the integral from -1 to 0 of 10/(x^2 + 1) is approximately 7.854. This integral represents the area under the curve f(x) from x = -1 to x = 0.
By using the Fundamental Theorem of Calculus, we know that the integral from a to b of f(x)dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
Since we have the integral value and we know the antiderivative of 10/(x^2 + 1) is arctan(x), we can write:
F(0) - F(-1) = 7.854
So, by plugging in the values, we get:
arctan(0) - arctan(-1) = 7.854
0 - (-π/4) = 7.854
π/4 = 7.854
Therefore, the integral from -1 to 0 of f(x) is π/4, which equals f(0) - f(-1). Since we know f(-1) = 8 (from the given point), we can solve for f(0):
f(0) - 8 = π/4
f(0) = 8 + π/4
f(0) ≈ 8.785
So, the estimated value of f(0) using integral calculations is approximately 8.785, not 13.
It seems there was an error in part (b) where you found f(0) = 13. Instead, the correct estimate for f(0) should be approximately 8.785 as calculated in part (c).
Keep in mind that these are approximations, and the actual values may differ depending on the level of accuracy required and the numerical methods used for calculation.