<p>I’ll take a stab at the points:</p>
<p>1.(a) 2 points: 1 - answer, 1 - units
(b) 2 points: 1 - interpretation of the integral, 1 - value of the integral
(c) 2 points: 1 - turning around at t = 2, 1 - justification
(d) 3 points: 1 - calculating Caren’s distance from school, 1 - calculating Larry’s distance from school, 1 - comparison of who lives closer</p>
<p>NOTE: Part (d) seems like it’s worth too many points (the comparison isn’t really calculus), but I’m not sure what other part could receive a third point instead.</p>
<p>2.(a) 2 points: 1 - integral, 1 - numerical solution
(b) 3 points: 1 - sets R’(t) = 0, 1 - finds the t-value, 1 - justifies an absolute maximum
(c) 2 points: 1 - connects w(2) - w(1) to integral(w’(t), t, 1, 2), 1 - numerical solution
(d) 2 points: 1 - integral, 1 - numerical solution</p>
<p>NOTE: I’ve sometimes seen problems like this before where the constant in front of the integral in part (d), in this case 1/(solution in part (a)) is part of the numerical solution point and not the integrand point.</p>
<p>3.(a) 2 points: 1 - calculates the cost to produce with an integral, 1 - profit
(b) 1 point - proper explanation that includes units
(c) 2 points: 1 - integral expression, 1 - rest of equation
(d) 4 points: 1 - finds P’, 1 - solves P’ = 0, 1 - evaluates P’ at student’s local max, 1 - justification (0/4 for any process to find P’ that doesn’t work with an integral expression)</p>
<p>4.(a) 3 points: 1 - limits, 1 - integrand, 1 - solution
(b) 3 points: 1 - integral, 1 - antiderivative of A(x), 1 - solution
(c) 3 points: 1 - limits, 2 - integrand as a function of y</p>
<p>NOTE: I’m not confident in the points for part (b) of this. Is it possible there’s another point elsewhere? I doubt they’ll regive the limits point for recycling the limits from part (a), but maybe?</p>
<p>5.(a) 2 points: 1 - difference quotient, 1 - solution
(b) 1 point: solution supported by student work
(c) 2 points: 1 - left Riemann sum set-up, 1 - solution
(d) 4 points: 1 - finds tangent line, 1 - finds secant line, 1 - evaluates tangent and secant lines at x = 7, 1 - uses concavity to justify inequalities</p>
<p>NOTE: Part of me wants to give part (b) 2 points and I’m not exactly sure what you could show as a 1 point answer. If it has 2 points, we probably remove a point from part (d), probably the third of those four points. I think there’s also a chance that the justification itself actually is two of the points for part (d), but it seems doubtful if all they want is a mention of the concavity. Maybe they want a better connection than that?</p>
<p>6.(a) 3 points: 1 - f "(x), 1 - identifies inflection points at x = 0 and x = 2, 1 - justification
(b) 4 points: 2 - f(-4), 2 - f(4)
(c) 2 points: 1 - x-value, 1 - justification</p>
<p>NOTE: I want to say that the points in (b) are broken down as 1 - calculates integral(f '(x), x, -4, 0), 1 - calculates integral(f '(x), x, 0, 4), 1 - antiderivative of f '(x) on [0, 4], 1 - uses initial condition of f(0) = 5 properly. Not sure what they’ll do there exactly. Pretty confident in the overall part breakdowns for this one.</p>
<p>Nitpick away and offer suggestions/alternatives!</p>