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Flashcards · AP® Calculus BC · Unit 4

Unit 4: Contextual Applications of Differentiation flashcards

45 cards on topics 4.1 to 4.7. Pick a topic or study the whole unit.

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4.1 Interpreting the Meaning of the Derivative in Context

What does the sign of f′(a) tell you in context?
f′(a) > 0: the quantity is increasing at a. f′(a) less than 0: decreasing. f′(a) = 0: momentarily not changing.
What are the units of f′ and f″?
f′: units of f ÷ units of input. f″: units of f ÷ (units of input)². If W is litres and t is minutes, W′ is litres per minute and W″ is litres per minute per minute.
What four parts does a derivative interpretation sentence need?
The quantity, the input value with units, increasing or decreasing, and the rate with units. Example: "At t = 10 minutes, the amount of water in the tank is increasing at a rate of 3 litres per minute."
Why is "decreasing at a rate of −5" wrong?
It is a double negative. Either say "decreasing at a rate of 5 …" or "changing at a rate of −5 …".
Does f′(a) give the exact change over the next unit of input?
No. f′(a) × Δt only approximates the change over a short step, because the rate itself may change.
h(t) is a plant's height in cm, t in days, and h′(10) = 0.8. Interpret it.
On day 10 the plant is growing at 0.8 cm per day (8 mm per day).
W(4) = 500 litres and W′(4) = −20 litres per hour. Is the tank empty soon after t = 4?
No. It is losing water at 20 litres per hour at that instant, but it still holds about 500 litres. An amount and a rate are different things.

4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration

How are position, velocity, acceleration and speed related?
v(t) = x′(t), a(t) = v′(t) = x″(t), and speed = |v(t)|. With metres and seconds: v in m/s, a in m/s².
How do you tell whether a particle is speeding up or slowing down?
Compare the signs of v and a. Same sign: speeding up. Opposite signs: slowing down. Negative acceleration alone does not mean slowing down.
When does a particle change direction?
Only where v changes sign. v = 0 alone is not enough: check the sign of v on each side. Turning points come from v, not from a = 0.
x(t) = t² − 6t + 5. Find v(1), a(1) and the speed at t = 1. Is the particle speeding up?
v = 2t − 6, so v(1) = −4; a = 2; speed 4. v and a have opposite signs, so it is slowing down.
How do you find total distance travelled from a position function?
Find every time where v changes sign in the interval, then add |change in x| over each piece between turning points. |final − initial| is the displacement only.
What does a velocity graph show directly?
Its slope is the acceleration, and its distance from the t-axis is the speed. Where it has a corner, a(t) does not exist.

4.3 Rates of Change in Applied Contexts Other Than Motion

Does a derivative have to be a rate with respect to time?
No. The input can be altitude, distance, price or number of items. f′ is a rate in any setting: dollars per item, grams per centimetre, people per year.
What does f′ less than 0 with f″ > 0 mean in context?
The quantity is decreasing, but more slowly. Similarly f′ > 0 with f″ less than 0 means increasing, but more slowly.
N(t) = 40 + 6t − t² visitors, t in hours. Find N′(2) and interpret it.
N′(t) = 6 − 2t, so N′(2) = 2: at t = 2 hours, the number of visitors is increasing at 2 visitors per hour.
In the visitor model N″(t) = −2. What does this mean at t = 2?
The number of visitors is still increasing, but the rate of increase is falling by 2 visitors per hour each hour. f″ less than 0 means the rate is falling, not the quantity.
R(q) is revenue in euros from selling q tickets. What are the units of R′(q)?
Euros per ticket. Check units are the right way up (output per input).
How do you estimate f′(c) from a table?
f′(c) ≈ (f(b) − f(a))/(b − a), using the closest values a and b on either side of c. This assumes the function is smooth between table values.

4.4 Introduction to Related Rates

What is the basis of related rates?
Quantities linked by an equation change together. Differentiate the equation with respect to time t; the chain rule gives d/dt g(x) = g′(x)·dx/dt.
What are d/dt of x², xy and πr²h when both r and h change?
d/dt x² = 2x·dx/dt; d/dt (xy) = x·dy/dt + y·dx/dt; d/dt (πr²h) = 2πrh·dr/dt + πr²·dh/dt.
In what order do you work a related-rates problem?
(1) Write an equation true at all times; (2) differentiate with respect to t; (3) substitute the values at the instant; (4) solve and give units. Substituting first turns rates into 0.
A = πr², r = 5 cm and dr/dt = 2 cm/s. Find dA/dt.
dA/dt = 2πr·dr/dt = 2π(5)(2) = 20π cm²/s.
P = xy, x = 2, y = 7, dx/dt = 3 and dy/dt = −1. Find dP/dt.
x·dy/dt + y·dx/dt = 2(−1) + 7(3) = 19.
What sign does a shrinking quantity's rate have?
Negative: "shrinking at 2 cm/s" means a rate of −2 cm/s. Also convert units (metres and centimetres) before substituting.

4.5 Solving Related Rates Problems

What are the six steps for a related-rates problem?
(1) Draw and label; (2) list known rates with signs, the instant and the rate wanted; (3) write an equation true at all times; (4) remove any extra variable; (5) differentiate with respect to t, then substitute; (6) answer with units and interpret.
Which quantities get letters and which get numbers in a related-rates equation?
Letters for anything that changes; numbers only for quantities that never change. A value "at the moment when…" is not a constant.
A right triangle has one leg fixed at 12 and the other leg x increasing at 2 units/s. How fast is the hypotenuse changing when x = 9?
x² + 144 = D², so x·dx/dt = D·dD/dt. When x = 9, D = 15, so dD/dt = (9/15)(2) = 1.2 units/s.
How do you handle a cone-shaped tank, point down, with rim radius R and depth H?
Use similar triangles r = (R/H)h to remove r, so V = (1/3)π(R/H)²h³ and dV/dt = π(R/H)²h²·dh/dt. With R = 2, H = 8: V = πh³/48.
What is the related-rates equation for an angle θ seen from a fixed point at adjacent distance k?
tan θ = x/k, so sec²θ·dθ/dt = (1/k)·dx/dt. Do not miss sec²θ, and work in radians.
What is a quick fact for the rate a liquid level rises in a container?
dh/dt = (dV/dt) ÷ (area of the liquid surface).
How should a related-rates answer be stated?
Say what changes, the direction, the size with units, and the instant. Example: "When the depth is 3 m, the depth is decreasing at 0.5 metres per minute."

4.6 Approximating Values of a Function Using Local Linearity and Linearization

What is the linearization of f at x = a?
L(x) = f(a) + f′(a)(x − a), the tangent line at a. Use f(x) ≈ L(x) for x close to a.
f(1) = 4 and f′(1) = −3. Estimate f(1.1).
L(1.1) = 4 − 3(0.1) = 3.7.
How do you decide whether a tangent-line estimate is an over- or underestimate?
Use f″ (the bend) between a and x. f″ > 0 (bends upward): tangent below the curve, underestimate. f″ less than 0 (bends downward): tangent above, overestimate. The sign of f′ does not decide it.
Estimate √49.7 using a = 49. Is it an over- or underestimate?
f′(x) = 1/(2√x), so f′(49) = 1/14. L = 7 + (1/14)(0.7) = 7.05. √x bends downward, so 7.05 is an overestimate (actual ≈ 7.0498).
How do you choose the point a for a linear approximation?
Choose a close to the x you want where f(a) and f′(a) are easy: perfect squares for √x, 8 or 27 for ∛x. The error grows as x moves away from a.
What mistakes are common with linearization?
Using f′(x) or f′ at the wrong point; dropping f(a); using x instead of (x − a); swapping over and under; writing = instead of ≈.

4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms

What does L'Hospital's Rule state?
If lim f(x) = 0 and lim g(x) = 0 (or both are infinite), then lim f(x)/g(x) = lim f′(x)/g′(x), provided the new limit exists or is infinite. It works for x → a, one-sided limits and x → ±∞.
What written check must come before using L'Hospital's Rule?
"lim (x → a) f(x) = 0 and lim (x → a) g(x) = 0, so L'Hospital's Rule applies" (or both infinite). Repeat the check each time you apply the rule again.
Find lim (x → 0) (sin 4x)/x.
4. The form is 0/0, and the rule gives lim 4 cos 4x/1 = 4.
Find lim (x → ∞) x³/eˣ.
0. It is ∞/∞; apply the rule three times: 3x²/eˣ, 6x/eˣ, 6/eˣ → 0.
Can you use L'Hospital's Rule on lim (x → 0⁺) (cos x)/x?
No. The top tends to 1, not 0, so it is 1/0: unbounded, tending to +∞. Using the rule on a non-indeterminate limit gives wrong answers.
What mistakes are common with L'Hospital's Rule?
Skipping the check; using the quotient rule on f/g instead of differentiating top and bottom separately; writing "= 0/0" as a value; forgetting the chain rule (d/dx sin 5x = 5 cos 5x).
f(a) = g(a) = 0, the derivatives are continuous and g′(a) ≠ 0. What is lim (x → a) f(x)/g(x)?
f′(a)/g′(a), by L'Hospital's Rule.

Every card in this unit is shared with the AP® Calculus AB flashcards, because the two courses share these topics.

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