Exam (elaborations) Calculus I MATH 1512 (4)

1. If (c, f(c)) is a point of inflection of the graph of f, then either f"(c)=0 or f" is undefined at c Answer: Points of Inflection 2. Where first derivative is 0 or undefined Answer: Critical Value 3. Use critical values and END POINTS in the function Answer: Find absolute extrema 4. If mn: NO HA If m=n: HA = co-eff of m/co-eff of n If mn: HA: y = 0 Answer: Horizontal Asymptote Rules 5. If f(x) is continuous on [a,b] and differentiable on (a,b), there is at least one point (x=c) where f'(c)= F(b)-F(a)/b-a Answer: Mean Value Theorem 6. If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occurs at critical points in the interval or at endpoints of the interval. Answer: Extreme Value Theorem 7. If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k Answer: Intermediate Value Theorem 8. If f(x) is continuous on the closed interval [a, b], differentiable on (a, b), and satisfies f(a) = f(b), then for some c in the interval (a, b), we have f'(c) = 0 Answer: Rolle's Theorem 9. Find HA: y = (x+2)/(sqrt(x^2+3)) Answer: y= +1 and y = -1 10.A pair of equations that define the x and y coordinates of a point in terms of a third variable called a parameter. Answer: parametric equations 11. Given x and y, how to parameterize? Answer: Table: |t | x | y| Plug in x,y to graph 12.An object moving along a line through the point (x0, y0), with dx/dt = a and dy/dt = b, has parametric equations Answer: x = x0 + at, y = y0 + bt 13.Horizontal Asymptote for exponential functions? Answer: y = k(if y = e^x + k) 14.If the function is not continuous, does the limit exist? Answer: The limit exists if there is removable discontinuity 15.What happens if you half delta(n) while calculating integral Answer: The difference between upper and lower estimate gets halved; more accurate prediction since velocity is measured more frequently