Calculus Questions - Optimization and another...

[blackout621]

Hey all, I'm really struggling with a couple of Calculus problems. If someone could walk me through to the correct answer for one or both of them, that would be much appreciated. I'd love to actually understand this stuff.

The above graph is g'(x).

If g(-2) = 1, what are the possible values for g(0)? This problem is looking for an interval(s) of answers.
g(0) is in _____.



Consider a window the shape of which a rectangle of height h surmounted by a triangle having a height T that is 1.3 times the width W of the rectangle.

If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.

H =

W =


These problems have really got me stumped... help is much appreciated. Thank you in advance!

 

[stapel]

The above graph is g'(x).

If g(-2) = 1, what are the possible values for g(0)? This problem is looking for an interval(s) of answers.

The value of the function is 1 when x = -2. The derivative, at x = -2, looks to be at or close to zero. After x = -2, the derivative is positive. What does this say about whether the original function is increasing, decreasing, or neither? In particular, is the value of g(x) going up, going down, or staying the same on the interval (-2, 0)?

Eye-balling, I think the max point of the derivative is at around (1, 1). What seems to you to be the maximum possible value of g'(x) on the interval (-2, 0)? In other words, how great might the slope be at some point on that interval? Given that value of m and the initial value g(-2) = 1, what line equation can you form? What is its value at x = 0? What can you conclude from this?

Consider a window the shape of which a rectangle of height h surmounted by a triangle having a height T that is 1.3 times the width W of the rectangle.

If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.

You have a rectangle with height h and width w. You have an isosceles triangle with base b = w and height T = 1.3w. What expression then stands for the total cross-sectional area?

Given the base and the height of the triangle, what does the Pythagorean Theorem tell you about the lengths of the slanty sides of the triangle? What expression then stands for the perimeter?

Given that the cross-sectional area is given as being A, solve the area equation for one of the variables. (The value of A is a constant; it's an unknown constant, but it is constant, not variable.) Plug this into the perimeter expression. Set equal to P, forming an equation for the perimeter P in terms of whichever one variable you have left. Minimize.