Solution for QID #21861: 1) Evaluate the following limits. a. limx2(1/x-2-12/x3-8) | StudyHelpMe

1) Evaluate the following limits. a. limx2(1/x-2-12/x3-8) b lim x-∞2+x/4+2x c lim –o (1/x1+x-1/x) 2. Use the limit definition of the derivative to show the following. 3 a) If f(x) = ——, then f’(1) = -} [3] (a) f(z) = 5; then fl) = —4 1 3 b) If f(z) = Vx +1, tl “2) = —— 3 (b) If fle) = VEFI, then f'(2) = 7 3. Find the derivative of the following functions. 3 (a) f(x) = ex2+x-e2x/x2-1 . r) —e€ __* Vx? -—1 xl/? + In(2x) b ©) = ———_—_ 4. The curve with equation 2x5 + 5y3 = 7 passes through the point (1, 1). [2] (a) Use implicit differentiation to find = at (1,1). [2] (b) Find the equation of the line which is tangent to the curve at (1,1). [4] 5. Suppose the volume of a sphere is expanding at the rate of 1.5 (m* per minute). How quickly is its surface area increasing at the moment when its volume is 4a (m*)? , ; a 4nr® Hint: For a sphere of radius 7, the volume is given by V = a and the surface area is given by S = 4zr?. 6. Let f(x) = e7* —e7**, [1] (a) What is the domain of f . [1] (b) Find the asymptote(s) of the function, if exists . [1] (c) Find the x- and y- intercepts of y = f(z) . [2] (d) Determine lim f(z). z—+00 [1] (e) On which intervals is f(x) increasing or decreasing ? [1] (f) On which intervals is f(x) concave up or down ?   [3] (g) Use the above parts to sketch the graph of the function. Label all critical points, points of inflections, x- and y- intercepts, local minima and maxima, and asymptotes, if exist.   [4] 7. Use the linear approximation to show \/1 + /z = zy   Hint: Use f(z) = /1+ 2, with a = 4. 8. Evaluate the following integrals (show all of your works). [3] (a) f In(x) dx 1 2 3 [3] (b) [ xe” dr 0 [6] 9. Determine the length of the curve y(x) = 42? — 3 In(x) on the interval [1, 2]. [6] 10. A bacterial population starts with 150 bacteria. Its growth rate N(t) varies with respect to time ¢. If the growth rate N(t) equals 300 e” (bacteria/hour), how many bacteria will there be after three hours?