ot . Consider the function defined by f(z) =In(y/|1 z2|). e (1 mark) State the domain of this function. (2 marks) Determine the derivative of In(,/|1 z2|). Be careful to show all your work. . (3 marks) Evaluate lim 2 sin(x) sin(2x) z0 2e* 2 2z 2 . (4 marks) A farmer wants to fence in a rectangular field. The farmer must use a different type of fencing for each pair of parallel sides of the field. One type costs $10 per metre, and the other type costs $5 per metre. Assuming the farmer can spend $800 on fencing in total, what are the dimensions of the largest possible field she could make? . (3 marks) Use logarithmic differentiation to find a formula for f'(x) if f(z) = 25" (2 1)?(322 4-2)%. . (4 marks) Find the point on the parabola z? = 2y that is closest to the point (4, 1). To find the domain of the function f(x) = In(v'1 x?), we need to consider the restrictions on the expression inside the logarithm and the square root functions. 2 must be Inside the square root, 1 greater than or equal to zero to ensure the square root is defined. So,1 z > 0. Inside the logarithm, the argument must be greater than zero. Thus, V1 22 > 0. Solving the inequality 1 r? > 0, we get: 1~ 2 Now, to find the derivative of f(x), let's differentiate step by step using the chain rule and the properties of logarithms: f(z) = In(VI2?) First, differentiate the inside function: g(z) = v1-a? d(z)=z(V1-2?) = ;~ %(1 z%) = (C2) = 5 Then, apply the derivative of the natural logarithm: f,(ZC) - 11 . 1_xx2 = 1-2c2 So, the derivative of f (x ) = In(V1 - x2 ) is f' ( ac - 20 1-2 2 .To evaluate the limit: . 2 sin(x)sin(2x) lim, 2e2-2% We'll use L'Hbpital's Rule, which states that if the of the ratio of two functions % as x approache . . B . certain value is of the form % or , then the lirr the ratio of their derivatives is the same as the original limit. Let's differentiate the numerator and the denomi separately: Numerator: f(x) = 2 sin(x) sin(2x) f (x) = 2 cos(x) 2 cos(2x) Denominator: g(x)=2" 2 2x x* g(x) =2"22x Now, let's evaluate the limit of the ratio of the derivatives as x approaches 0O: f(x) _ 2cos(0)-2cos(0) _ 2-2 _ 0 limy, 0 g (x) 20-2-2-0 220 Since we still have an indeterminate form 8, we apply L'Ho6pital's Rule again