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Question #1. Consider f(x) = v3x + 1 and h # 0. i. Compute the average f(x + h) - f(I) ii. Use the previous (simplified) average to compute f'(x) = lim f( x th) - f(x) h-+0o h iii. Use the obtained limit/derivative to compute f'(1). Question #2. Consider the function f(x) given in Figure 1. Determine the points in which f is not differentiable. y for -10 6 Figure 1: Graph of f(x). Question #3. Consider the following code for the rules of differentiation: D1. Sum/Subtraction: [f(x) +g(x)]' = f'(x)+9'(x). D5. Power: d [ak ] = kak - 1 . D2. Constant Multiple: [of (x)]' = of'(x). D3. Product: If (x)-g(x)]' = f'(x)-9(x)+f(x)-9'(x). D6. Constant: [c] = 0. D4. Quotient: f(x)]' f'(x)9(x) - f(x)9(2) g(2) [g(x)] D7. Chain Rule: If(g(x))]' = f'(g(x))g'(x). Let f(x) = -20 3-2v and determine which property was used in each step: f'(2) = 2-2Vz I [x3 - 2Vx]' . x - [x3 -2Vx] . 1 [x]2 ([x3]' - [2Vx]') . x- [23 -2Vx] (312 - 274 7 - [23-2Vx 213 + x Question #4. Compute f'(x) for: i. f(x) = [ + 4x + 3 iii. f(x) = 2 - tan(I) ii. f(x) = x1/5 - sin(x). iv. f(x) = (2x3 + 3)(x4 - 2x). Question #5. Let f(x) = -. Find f'(x) and f"(x). Question #6. Consider the curve given by y = Va r + 1. Find an equation for the tangent line to the graph at P = (4, 0.4). Question #7. Use the Chain Rule to differentiate: i. F(x) = ii. F(x) = (x2 - x+1)3. V12 + 4 ddy du Question #8. Find du' dx and d i. y = u2/3 and u = 1+ 24. ii. y = cos(u) and u = 3x2 + 1. Question #8. Find y' and y" where y is given implicitly by i. 9x2 + y? = 9. ii. Vi + Vy = 1