for the curve y=(x^2+1)/(x^2-4), find:
(i). The cordinates of the turning points
(ii). The equations of the asymptotes
(iii). Sketch the curve
dy/dx = (2x(x^2 -4) - 2x(x^2+1))/(x^2-4)^2
= (2x^3 - 8x - 2x^3 - 2x)/(x^2-4)^2
= -6x/(x^2-4)^2
= 0 for max/min or turning points
6x/(x^2-4)^2 = 0 ---> x = 0
then y = -1/4
the turning point is (0, -1/4)
asymtotes .... when the denominator is zero
x^2 - 4 = 0
x = ± 2
VA at x = 2 and x = -2
http://www.wolframalpha.com/input/?i=plot+y%3D%28x%5E2%2B1%29%2F%28x%5E2-4%29
To find the coordinates of the turning points of the curve y = (x^2+1)/(x^2-4), you'll need to find the derivative of the function and solve for x when the derivative equals zero.
(i) Find the derivative:
To find the derivative of the given function y = (x^2+1)/(x^2-4), use the quotient rule. The quotient rule states:
If y = f(x)/g(x), then y' = (f'(x)g(x) - g'(x)f(x))/(g(x))^2.
Let's apply the quotient rule to find the derivative of y:
f(x) = x^2+1
g(x) = x^2-4
Calculating the derivatives:
f'(x) = 2x
g'(x) = 2x
Now apply the quotient rule:
y' = [(2x)(x^2-4) - (2x)(x^2+1)] / (x^2-4)^2
Simplify the equation further:
y' = -6(x^2 + 2) / (x^2 - 4)^2
(ii) Find the coordinates of the turning points:
To find the turning points, set the derivative equal to zero and solve for x:
-6(x^2 + 2) / (x^2 - 4)^2 = 0
Since the numerator is zero, we have:
x^2 + 2 = 0
Solving for x, we get:
x^2 = -2
x = ±√(-2)
Since we have a square root of a negative number, it means that there are no real turning points for this function since the denominator is always positive.
(iii) Find the equations of the asymptotes:
To find the asymptotes, we need to analyze the behavior of the function as x approaches positive or negative infinity.
As x approaches positive or negative infinity, the values of the function y will approach certain limits. There are three possibilities: horizontal asymptote, vertical asymptote, or oblique (slant) asymptote.
1. Horizontal asymptote:
To find the horizontal asymptote, compare the degrees of the highest powers of x in the numerator and denominator. In this case, the degree of the numerator and denominator is both 2. If the degrees are the same, divide the coefficients of the highest power. The horizontal asymptote is therefore given by the ratio of the leading coefficients:
y = (1/1) = 1
Thus, the equation of the horizontal asymptote is y = 1.
2. Vertical asymptotes:
To find the vertical asymptotes, set the denominator equal to zero and solve for x:
x^2 - 4 = 0
x^2 = 4
We have two possible values for x: x = 2 and x = -2
Thus, the equations of the vertical asymptotes are x = 2 and x = -2.
Now let's sketch the curve:
To sketch the curve, we've already determined the vertical asymptotes and the horizontal asymptote.
(i) Horizontal asymptote: y = 1
(ii) Vertical asymptotes: x = 2 and x = -2
Plot the vertical asymptotes on the x-axis at x = 2 and x = -2.
Since the function has no real turning points, you may plot some points on the curve by choosing various x-values and calculating the corresponding y-values using the given function equation.
Connect the points and sketch the curve following any trends or patterns you observe.
Note: Without a scale or exact values for the points, it's difficult to provide an accurate sketch. I encourage you to plot the curve on graph paper or using software that can graph functions.