Hi there,could you help me with the following questions;

1.) Analyze the key features (any asymptotes, intercepts, critical points) and then sketch the graph of the function f(x)= x-0.25/x^2

--> I am unsure about my answers could you tell me if I am correct;
Intercepts: x=0.25 y=0
Asymptotes: Vertical (x=0)/ Horizontal (x=0) f(100)= >0 above and f(-100)= <0 below
Critical points: x=0 and x=0.5

2.) A custom designed large window has the shape of a rectangle with a semicircle added on the top. To ensure adequate illumination, the area of the window needs to be 12m^2. Find the dimension of the window (base and height of the rectangle) if the perimeter is a minimum.
Find the minimum perimeter of the window. They also tell you the base of the rectangle (2r) and the height (h). We can also assume the base of the rectangle (2r) is the base of the semicircle also.

--> I am confused with this question.

Sure, I can help you with the questions.

1.) Analyzing the key features of the function f(x) = (x - 0.25) / x^2:

To analyze the key features of the function, we can start by finding the intercepts, asymptotes, and critical points.

Intercepts: To find the x-intercept, we set f(x) = 0 and solve for x. In this case, (x - 0.25) / x^2 = 0. From this equation, we can see that the numerator, (x - 0.25), must be equal to 0 for f(x) to be equal to 0. Therefore, x = 0.25 is the x-intercept. The y-intercept is the value of f(x) when x = 0, which can be found by substituting x = 0 into the equation. In this case, f(0) = (0 - 0.25) / 0^2 = undefined. So, there is no y-intercept.

Asymptotes: To find the vertical asymptote, we look for values of x that make the denominator (x^2) equal to 0. In this case, the denominator is never equal to 0. Therefore, there is no vertical asymptote. For horizontal asymptotes, we look at the behavior of the function as x approaches positive or negative infinity. As x becomes large (positive or negative), the leading term in the function is x^(-2), which tends to 0. Therefore, the horizontal asymptote is y = 0.

Critical points: To find critical points, we calculate the derivative of the function and set it equal to 0. Differentiating f(x), we get f'(x) = (1 * x^2 - (x - 0.25) * 2x) / x^4. Setting f'(x) = 0 and solving for x, we have x^2 - 2x^2 + 0.5x = 0, which simplifies to -x^2 + 0.5x = 0. Factoring out x, we get x(-x + 0.5) = 0. This equation has two solutions: x = 0 and x = 0.5. Therefore, x = 0 and x = 0.5 are the critical points.

Sketching the graph: Based on the key features we have determined, we can now sketch the graph of the function f(x) = (x - 0.25) / x^2. The graph will have an x-intercept at x = 0.25 and a horizontal asymptote at y = 0. Since there is no vertical asymptote, the graph will not have a vertical line that it approaches or touches. The critical points at x = 0 and x = 0.5 can be identified on the graph as well. It is important to note that we also need to consider the behavior of the function between the key points to accurately sketch the graph.

2.) The custom-designed large window is in the shape of a rectangle with a semicircle added on the top. The area of the window needs to be 12m^2, and we need to find the dimensions (base and height of the rectangle) that minimize the perimeter.

To solve this problem, we can use the concept of calculus optimization. Let's denote the base of the rectangle as 2r and the height as h. The area of the rectangle can be expressed as A = 2rh, and the area of the semicircle as Ac = πr^2/2. The total area of the window is given as 12m^2, so we have the equation 2rh + πr^2/2 = 12.

To find the dimensions that minimize the perimeter, we need to minimize the perimeter formula P = 2r + 2h + πr = 2(r + h) + πr. Since we want to minimize the perimeter, we should minimize the expression (r + h) + (π/2)r.

We are given that the base of the rectangle is 2r, so we can substitute r = (base/2) into the equation for the area. This gives us 2h(base/2) + π(base/2)^2/2 = 12. Simplifying this equation, we get hb + (π/4)b^2 = 12.

Now, we can express h in terms of b using the equation hb + (π/4)b^2 = 12. Rearranging the terms, we have hb = 12 - (π/4)b^2. Dividing both sides by b, we get h = (12 - (π/4)b^2)/b.

Substituting this expression for h into the expression to minimize the perimeter, we have P = 2(r + h) + πr = 2((base/2) + (12 - (π/4)b^2)/b) + π(base/2). Simplifying further, we have P = base + (12 - (π/2)b^2/b) + π(base/2). P = base + (12 - (π/2)b) + (π/2)base.

To find the dimension of the window (base and height) that minimizes the perimeter, we can differentiate P with respect to b, set it equal to 0, and solve for b. After finding the value of b, we can then substitute it back into the equation to find the corresponding base and height.

I hope this helps! Let me know if you have any more questions.