1. Stating the problem: We analyze the differentiability and continuity of the functions graphed in Exercises 43-48 over their given domains. 2. Important concepts: - A function is differentiable at a point if it has a defined tangent (smooth slope) there. - A function is continuous at a point if there is no break, jump, or hole. - Points where the function is continuous but not differentiable include sharp corners or cusps. - Points where the function is neither continuous nor differentiable include jumps or holes. 3. Exercise 43: Linear function from (-3,2) to (2,-2) - Linear functions are differentiable and continuous everywhere on their domain. - So, differentiable on [-3,2], continuous on [-3,2], no exceptions. 4. Exercise 44: Wavy curve from (-2,0) to (3,1) - Smooth curve with no sharp corners or breaks. - Differentiable and continuous on entire domain [-2,3]. 5. Exercise 45: Two parts with open circles at (-1,1.5), (0,0), and (3,0) - Open circles indicate points not included, so discontinuities at x=-1,0,3. - At these points, function is neither continuous nor differentiable. - Elsewhere, smooth curves imply differentiability. 6. Exercise 46: Sharp V-shaped point near (-1,0), open circle at (1,3) - Sharp V at x=-1 means continuous but not differentiable there. - Open circle at x=1 means discontinuity, so neither continuous nor differentiable at x=1. - Elsewhere differentiable. 7. Exercise 47: Smooth curve from (-1,1.25) to (2,1.5) - No sharp corners or breaks. - Differentiable and continuous on [-1,2]. 8. Exercise 48: Two sharp peaks at about (-2,4) and (2,3), valley near (0,0) - Sharp peaks mean continuous but not differentiable at x=-2 and x=2. - Valley is smooth, so differentiable there. - Continuous everywhere on [-3,3]. Final answers: - Differentiable points: 43: all [-3,2] 44: all [-2,3] 45: all except x=-1,0,3 46: all except x=-1 (sharp corner), x=1 (open circle) 47: all [-1,2] 48: all except x=-2,2 (sharp peaks) - Continuous but not differentiable: 46: x=-1 48: x=-2,2 - Neither continuous nor differentiable: 45: x=-1,0,3 46: x=1