Self-check: 201-103, Differential Calculus for Social Science
One question per chapter, 12 in all, taken from the course's corrected exercises. Answer the ones you can do and skip those from a chapter you have not seen yet: each answer is checked right away, and the solution opens after.
With your free account, each answer is kept and your dashboard files each chapter: passed the check, or to work on. These are the same questions as on the set pages: an answer given here counts there.
Fee schedules, rates and aid grids are almost all PIECEWISE-defined functions: one formula per bracket, and a joint between two brackets.
A mobile data plan costs C(x) dollars per month for x gigabytes used: C(x)=45 for 0≤x≤20, C(x)=45+7(x−20) for 20<x≤50, and C(x)=250 for x>50, the plan becoming unlimited.
a) Calculate C(12), C(35) and C(60).
See the solution
Answer
a)C(12)=45, C(35)=150, C(60)=250 dollars.
a) x=12 falls in the first bracket: C(12)=45 dollars. x=35 falls in the second: C(35)=45+7×(35−20)=45+105=150 dollars. x=60 falls in the third: C(60)=250 dollars. The one move to take care over is to READ the condition before choosing the formula; applying the second formula to x=60 would give 45+280=325 dollars, an amount nobody pays.
A limit does not ask what the function equals at a point: it asks where the function is HEADING as you approach that point. The two questions often have the same answer, and the whole chapter lives in the cases where they do not.
The graph shows a function f defined on [0,9]. A filled dot marks a value that is attained, an open dot a value that is approached but not attained.
a) Give limx→2−f(x), limx→2+f(x), limx→2f(x) and f(2).
See the solution
Answer
a)limx→2−f=limx→2+f=5, so limx→2f=5, whereas f(2)=3.
a) Approaching 2 from the LEFT, we follow the branch x+3, which tends to 2+3=5. Approaching from the RIGHT, we follow the branch 7−x, which tends to 7−2=5. The two one-sided limits are equal, so limx→2f(x)=5. But the filled dot is at (2,3): f(2)=3. The limit exists and equals 5, the value exists and equals 3, and they differ. This is the case that on its own justifies the invention of the limit: the function knows where it is going without being there.
Chapter 3 · The derivative and differentiation rules
Once the definition is understood, it is no longer used: the rules do the work. Three of them cover most of the course, and two writing traps cost more marks than the rules themselves.
The first trap is to reverse the order of the numerator in the quotient rule; the second is to believe that the derivative of a product is the product of the derivatives.
a) Differentiate f(x)=5x4−x23+7x−9.
See the solution
Answer
a)f′(x)=20x3+x36+2x7.
a) First rewrite each term as a power: f(x)=5x4−3x−2+7x1/2−9. The rule (xn)′=nxn−1 then applies to all of them, negative and fractional exponents included. f′(x)=20x3+6x−3+27x−1/2, that is f′(x)=20x3+x36+2x7. Two signs to watch: −3x−2 gives −3×(−2)x−3=+6x−3, the minus becomes a plus, and the constant −9 disappears.
An annual rate of 2.3% means nothing to anyone; a doubling time of thirty years does. Going from one to the other takes one line of logarithm, and there is a mental rule that approximates it very well.
This is the rule that economists and demographers use in meetings, without a calculator.
e) An investment earns 20% per year, compounded ANNUALLY. Compare the exact calculation with the rule of 70 and explain the gap.
See the solution
Answer
e)ln1.2ln2≈3.80 against 3.5: the rule assumes ln(1+r)≈r, valid at small rates.
e) Here the compounding is ANNUAL, so the capital is C0×1.2t and not C0e0.2t. Doubling requires 1.2t=2, so t=ln1.2ln2=0.182320.69315≈3.80 years, whereas the rule announces 2070=3.5 years, an error of 8%. Two causes add up. First, the rule rests on the approximation ln(1+r)≈r, excellent for small r and mediocre for r=0.2, where ln1.2 is only 0.182 instead of 0.2. Second, interest paid once a year earns less than interest paid continuously. The rule of 70 is therefore a tool for small rates, which covers most of demography and macroeconomics, but not double-digit returns.
The whole mechanics of the chapter fits in a chain rule applied to a function you do not know. Every time a y goes under an operation, its derivative leaves a factor dxdy behind; every time x and y meet in the same term, the product rule speaks.
A household buys x units of good A and y units of good B. Both are delivered together, and the handling cost grows with the product of the quantities, so its spending constraint is 2x+3y+0.05xy=90 dollars. The current bundle is (15,16). The figure below shows the situation.
d) A student differentiates the term 0.05xy as 0.05y. What value does the student get, and by what percentage is it off?
See the solution
Answer
d)−0.933, that is 25 percent too much in absolute value.
d) By forgetting the second term of the product rule, the student writes 2+0.05y+3dxdy=0, so dxdy=−32.8≈−0.933. The ratio of the two values is 0.74670.9333=1.25: the student overstates the sacrifice by exactly 25 percent. An error like this does not show to the naked eye, because the result stays negative and plausible, and that is what makes it costly on an exam.
A population is a STOCK, fed and drained by FLOWS: births, deaths, migratory inflows and outflows. The derivative of a stock is the algebraic sum of its flows, and it is the simplest form of related rate.
A municipality has P=48,000 inhabitants. Over one year, it records 1,120 births, 860 deaths and a net migration of −140 people. The figure below shows the situation.
d) Births stay at 1,120 per year and the population grows as in a). Compute dtdN.
See the solution
Answer
d)dtdN≈−0.058 per thousand per year.
d) The numerator is constant and the denominator varies, so dtdN=1,000B×(−P21)dtdP=−48,00021,000×1,120×120=−2,304,000,000134,400,000≈−0.058 per thousand per year. The crude birth rate FALLS, although the number of births has not moved by a single unit.
Chapter 7 · Differentials and linear approximation
A social science datum is never exact: it is known to within a few percent. When you combine it with another, the uncertainties do not cancel, they add up, and the differential says exactly how.
The rule is the same as for the relative rates of chapter 5, applied to an error instead of a speed.
c) An area is estimated by A=πr2 from a radius known to within 3 %. What is the margin on A?
See the solution
Answer
c)AdA=2rdr, so 6 %.
c) AdA=2rdr, the exponent 2 coming out through logarithmic differentiation. The margin is therefore 2×3=6 %. A power MULTIPLIES the relative error by its exponent, which is why you never measure an area or a volume by measuring a length alone when you can do otherwise.
The logistic model describes anything that spreads through a closed population: a service, a practice, an idea. Its complete study comes down to four numbers, and the most useful of the four is not the ceiling.
The cumulative adoptions of a service are N(t)=1+29e−0.5t9,000, t in months.
c) The inflection point is reached when 29e−0.5t=1. Calculate its date and the corresponding y-coordinate.
See the solution
Answer
c)t=2ln29≈6.73 months, at 4,500 adoptions, that is, half the ceiling.
c) 29e−0.5t=1 gives e−0.5t=291, so −0.5t=−ln29 and t=2ln29≈2×3.367=6.73 months. The y-coordinate is then N=1+19,000=4,500, exactly HALF the ceiling. This is a general property of the logistic model: its inflection point is always at half height.
A training program produces a productivity gain with diminishing returns, and it costs the wages of the hours it takes up. The net benefit therefore has a maximum, and it arrives sooner than you would think.
The productivity gain of a training program of d days is 1,600d dollars per employee, and the opportunity cost is 200d dollars. The net benefit is N(d)=1,600d−200d.
e) A manager proposes doubling the length to double the effect. What do you answer, with figures to support it?
See the solution
Answer
e)N(32)≈2,651 dollars, which is 549 less: the gain follows 2, the cost doubles.
e) Doubling the length does not double the effect, because the gain follows a SQUARE ROOT: going from 16 to 32 days multiplies the gross gain by 2≈1.41 and not by 2, while the cost doubles exactly. The computation gives N(32)=1,60032−6,400≈9,051−6,400=2,651 dollars, which is 549 dollars LESS than with sixteen days. The proposal therefore loses money while tying up twice as much working time. The general rule, valid for any activity with diminishing returns: increase the effort as long as the MARGINAL gain exceeds the marginal cost, here as long as d800>200, and not an hour longer.
Chapter 10 · Trigonometric functions and their derivatives
At x=0, the quotient xsinx gives 00: an indeterminate form, just like those of the chapter on limits. Its value is neither 0 nor infinity, and it cannot be guessed.
It equals 1, and this is where ALL of the course's trigonometric differentiation comes from. The figure shows why: near zero, the sine curve and the line y=x become indistinguishable.
a) Compute 0.2sin(0.2) then 0.05sin(0.05), to four decimal places.
See the solution
Answer
a)0.9933 then 0.9996.
a) sin(0.2)=0.198669, so 0.2sin(0.2)=0.993347≈0.9933. Then sin(0.05)=0.0499792, so 0.05sin(0.05)=0.999583≈0.9996. The trend is clear: dividing the input by 4 divided the GAP FROM 1 by about 16. That is the signature of convergence in x2, and the approximation sinx≈x−6x3 confirms it, since it gives xsinx≈1−6x2. Careful: the calculator must be in radians, otherwise these two numbers are about 0.0175 and nothing converges to 1.
A firm does not sell at a fixed price: the more it produces, the more it has to lower its price to move its output. The demand function therefore links price to quantity, and the revenue follows from it.
Demand function: p=100−0.5q, where p is the unit price in dollars.
b) Compute the marginal revenue R′(q) and evaluate it at q=40.
See the solution
Answer
b)R′(q)=100−q, so R′(40)=60 dollars, and not the 80 dollars the customer pays.
b) R′(q)=100−q. At q=40: R′(40)=100−40=60 dollars. Selling a 41st unit would therefore bring in about 60 dollars more, not 80 dollars, even though 80 dollars is what the customer would pay.
A slope answers the question “by how much does the quantity fall if the price rises by one dollar?” Elasticity answers a different question, one with no unit: “by how many PERCENT does the quantity fall if the price rises by one percent?”
Demand function: q=200−2p, where q is the number of units and p the price in dollars.
d) The slope is constant and equals −2. Is the elasticity constant too? Comment.
See the solution
Answer
d)No: E equals 0 at p=0, −1 at p=50, and tends to infinity as p approaches 100.
d) No, the elasticity VARIES all along a demand that is nonetheless linear: it equals 0 at p=0, −1 at the midpoint, and tends to infinity as p approaches 100. The reason is in the formula: the slope is constant, but the factor qp is not, and it blows up as q becomes small. This distinction between slope and elasticity is the most misunderstood point of the chapter: a line does not have one elasticity, it has one per point.
Your dashboard files each chapter from your answers and tells you where to start. An exam date in mind? Note it on the course card: it builds a plan from it, day by day.
