1. In each of the following equations, suppose g is a given differentiable function of one...

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1. In each of the following equations, suppose g is a given differentiable function of one variable. Suppose the equation defines y implicitly as a function of x. Find an expression for y' in each case. (a) xy = g(x) + y (b) g(x + y) = x + y (c) (xy + 1) = g(xy) 2. Suppose the function f is defined for all real x by the following formula: f(x) = x + 3x + 6x - 3. Show that f has an inverse function g, and then, given that f(1)=7, find g(7). 3. Find the linear approximation to the following functions about x=0: (a) f(x) = (1+x) (b) f(x) = (1+x)5 (c) f(x) = (1 x)/4 4. Determine the values of x at which each of the functions defined by the following formulas is continuous: (a) x+1 (b) x8-3x+1 x+2x-2 (c) + (2+2)3/2 x 5. Use L'Hopital rule to find the following Limits: (a) lim1 In x-x+1 (x-1)2 xx-x (b) lim ln 7x+1 4x+4 (c) limx11x+ln x 6. Find the intervals where the following functions are increasing: (a) y = (lnx) 24 (b) y = ln (e* + ex) (c) y=x-In (x + 2) 7. What is the present value of 15 annual deposits of $3500 if the first deposit is after one year and the annual interest rate is 12%? 1. In each of the following equations, suppose g is a given differentiable function of one variable. Suppose the equation defines y implicitly as a function of x. Find an expression for y' in each case. (a) xy = g(x) + y (b) g(x + y) = x + y (c) (xy + 1) = g(xy) 2. Suppose the function f is defined for all real x by the following formula: f(x) = x + 3x + 6x - 3. Show that f has an inverse function g, and then, given that f(1)=7, find g(7). 3. Find the linear approximation to the following functions about x=0: (a) f(x) = (1+x) (b) f(x) = (1+x)5 (c) f(x) = (1 x)/4 4. Determine the values of x at which each of the functions defined by the following formulas is continuous: (a) x+1 (b) x8-3x+1 x+2x-2 (c) + (2+2)3/2 x 5. Use L'Hopital rule to find the following Limits: (a) lim1 In x-x+1 (x-1)2 xx-x (b) lim ln 7x+1 4x+4 (c) limx11x+ln x 6. Find the intervals where the following functions are increasing: (a) y = (lnx) 24 (b) y = ln (e* + ex) (c) y=x-In (x + 2) 7. What is the present value of 15 annual deposits of $3500 if the first deposit is after one year and the annual interest rate is 12%?