Unit 4 Beta

Contextual Applications of Differentiation: the one-page sheet

4.1 The Derivative in Context

In context, a derivative is a rate with units of output per unit of input. Interpret f′(a) by naming the instant, the quantity, whether it is increasing or decreasing, and how fast, with units. Keep amounts, f(a) , and rates, f′(a) , apart.

  • Units of f′f': units of ff divided by units of the input (liters per minute, dollars per item).
  • Interpret with four parts: the instant, the quantity by name, increasing or decreasing, and the rate with units.
  • f(a)f(a) is an amount; f′(a)f'(a) is a rate. Do not mix them in a sentence.
  • From a table, estimate f′(a)f'(a) with the two closest inputs on either side of aa, using the table's actual widths.

its units are output units per input unit says whether the quantity is increasing (positive) or decreasing (negative) at that instant says how fast, in those units at this instant, this quantity is increasing (or decreasing) at this many units of output per unit of input

derivative in context
The meaning of <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(a)</annotation></semantics></math> in a real situation: the rate at which the quantity <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> changes at input <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, in units of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> per unit of the input (for example, centimeters per hour).

4.2 Straight-Line Motion: Position, Velocity, Acceleration

For a particle on a line, v(t) = x′(t) and a(t) = v′(t) = x″(t) . The sign of v gives the direction, and the direction changes only where v changes sign. Speed is |v| ; it increases when v and a have the same sign and decreases when their signs differ.

  • Velocity has a sign; speed is |v(t)||v(t)| and is never negative.
  • The particle is at rest when v(t)=0v(t) = 0, and changes direction only where vv changes sign.
  • Speeding up when vv and aa have the same sign. State both signs in a justification.
  • Units: position in meters gives velocity in meters per second and acceleration in meters per second per second.

its rate of change is the velocity, v(t) = x′(t) its rate of change is the acceleration, a(t) = v′(t) = x″(t) gives the direction: right (or up) when v > 0 , left when v < 0 ; a direction change needs v to change sign same sign: speeding up; opposite signs: slowing down

position function
A function <math class="calc-m"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)</annotation></semantics></math> giving the location of an object moving along a line at time <math class="calc-m"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>, measured from a fixed origin, positive in one direction and negative in the other.
velocity
The rate of change of position: <math class="calc-m"><semantics><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi>x</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v(t) = x'(t)</annotation></semantics></math>. Its sign gives the direction of motion and its absolute value the speed. Units: distance per time.
acceleration
The rate of change of velocity: <math class="calc-m"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi>v</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi>x</mi><mo lspace="0" rspace="0">″</mo></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a(t) = v'(t) = x''(t)</annotation></semantics></math>. Units: distance per time per time, such as meters per second per second.
speed
The absolute value of velocity, <math class="calc-m"><semantics><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|v(t)|</annotation></semantics></math>: how fast an object moves, whatever its direction. Speed is never negative.
speeding up
A particle is speeding up when its speed <math class="calc-m"><semantics><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|v|</annotation></semantics></math> increases, which happens when <math class="calc-m"><semantics><mrow><mi>v</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v(t)</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a(t)</annotation></semantics></math> have the same sign; it is slowing down when they have opposite signs.

4.3 Rates of Change in Other Contexts

In any context, Q′ is the rate of change of Q in units of Q per input unit, and Q″ is the rate of change of that rate. Read the sign of Q′ for increasing or decreasing and the sign of Q″ for whether that rate grows or shrinks. Always state units.

  • An interpretation names the input value, the quantity, increasing or decreasing, and the rate with units.
  • When the function is itself a rate (liters per hour), its derivative is a rate of a rate (liters per hour per hour).
  • Q′<0Q' < 0 with Q″>0Q'' > 0: decreasing, but more slowly. Q′>0Q' > 0 with Q″<0Q'' < 0: increasing, but more slowly.
  • Chained quantities multiply rates, and their units cancel: (°F per foot)(feet per minute) = °F per minute.

has a rate of change Q′ in units of Q per unit of input says whether the quantity is increasing or decreasing measured in units of Q per input unit per input unit says whether the rate Q′ is increasing or decreasing

rate of change in context
The derivative of a quantity in a real setting other than motion (temperature, cost, population, volume): how fast the quantity changes per unit of input, with units, and with its sign telling whether the quantity is increasing or decreasing.

When quantities linked by an equation change with time, differentiate the equation with respect to t . The chain rule gives every changing variable its rate, producing an equation among the rates. Substitute the values of one instant afterward, with signs that match increasing or decreasing.

  • Related rates: differentiate an equation linking quantities with respect to time tt.
  • Each variable that changes gets its own rate, dxdt\frac{dx}{dt}, drdt\frac{dr}{dt}, from the chain rule. Constants get none.
  • A decreasing quantity has a negative rate: shrinking at 2 cm/s means drdt=−2\frac{dr}{dt} = -2.
  • Products need the product rule: ddt(xy)\frac{d}{dt}(xy)=dxdty+xdydt= \frac{dx}{dt}y + x\frac{dy}{dt}.

each one is a function of t , even if the equation shows no t the chain rule attaches a rate to each variable: d/dt r² = 2rdr/dt known rates and values give the unknown rate the rates are not lost to derivatives of constants

related rates
Rates of change of quantities that are linked by an equation. Differentiating the equation with respect to time <math class="calc-m"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> (chain rule on every changing variable) gives an equation linking their rates, such as <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>A</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>π</mi><mi>r</mi><mfrac><mrow><mi>d</mi><mi>r</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dA}{dt} = 2\pi r\frac{dr}{dt}</annotation></semantics></math>.

To solve a related rates problem: draw and label, write an equation linking the quantities at every instant, differentiate it with respect to time, then substitute the instant's values and solve for the unknown rate. Use geometry to reduce to one changing quantity, and give the answer with its sign and units.

  • The five steps: draw and label, relate, differentiate with respect to tt, substitute, answer with units.
  • Constants of the problem (a 10-foot ladder, a 6-foot person) stay numbers; quantities that change get letters.
  • Use similar triangles to remove an extra variable before differentiating, so only one unknown rate remains.
  • Shrinking or approaching means a negative rate. Report the answer in words too: "the top slides down at 1.5 ft/s".

each named quantity is a function of time t ; fixed lengths stay numbers eliminate any extra variable with a second relation first, such as r = h/2 the chain rule gives an equation among the rates solve for the unknown rate and state it with units

related rates problem
A problem that asks how fast one quantity changes, given how fast related quantities change: write an equation linking them, differentiate with respect to time, then substitute the values at the instant in question.

4.6 Local Linearity and Linearization

Near x = a , f(x) ≈ L(x) = f(a) + f′(a)(x − a) . The tangent line lies below a concave up graph ( f″ > 0 ), giving an underestimate, and above a concave down graph ( f″ < 0 ), giving an overestimate. The estimate is best close to a .

  • The linearization at aa: L(x)L(x)=f(a)+f′(a)(x−a)= f(a) + f'(a)(x - a). Use the point closest to where you estimate.
  • Concave up (f″>0f'' > 0): tangent line below the graph, underestimate. Concave down (f″<0f'' < 0): overestimate.
  • Justify with the concavity on the interval between aa and xx, not with the sign of f′f'.
  • The closer xx is to aa, the better the approximation.

it is close to its tangent line L(x) = f(a) + f′(a)(x − a) f(x) ≈ f(a) + f′(a)(x − a) , the tangent line approximation the approximation is an underestimate the approximation is an overestimate

overestimate
Whether an approximation is too large (an overestimate) or too small (an underestimate) compared with the true value. For a tangent line approximation it follows from concavity.
linearization
The tangent line used to approximate a function near a point: <math class="calc-m"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(x)</annotation></semantics></math><wbr><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= f(a) + f'(a)(x - a)</annotation></semantics></math>, so <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) \approx L(x)</annotation></semantics></math> for <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> near <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>. Also called local linear approximation or tangent line approximation.
concave up
A graph is concave up on an interval where <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'' &gt; 0</annotation></semantics></math> (it bends upward and lies above its tangent lines) and concave down where <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'' &lt; 0</annotation></semantics></math> (it bends downward and lies below them).

4.7 L'Hospital's Rule

For a limit of the form 0/0 or ∞/∞ , L'Hospital's rule gives lim (f(x))/(g(x)) = lim (f′(x))/(g′(x)) , derivatives taken separately. Show both limits first; repeat if the new limit is still indeterminate; and evaluate directly when the form is not indeterminate.

  • First show the form: write limf(x)=0\lim f(x) = 0 and limg(x)=0\lim g(x) = 0 (or both infinite). Without it the rule does not apply.
  • Differentiate the top and the bottom separately. It is not the quotient rule.
  • If the form is not indeterminate, just evaluate: L'Hospital's rule would give a wrong answer.
  • Never write 00\frac{0}{0} as a value with an equals sign; describe the form in words or with two limits.

plugging in tells nothing: the answer could be any number, or no limit at all (f(x))/(g(x)) ≈ (f′(a)(x − a))/(g′(a)(x − a)) = (f′(a))/(g′(a)) lim (f(x))/(g(x)) = lim (f′(x))/(g′(x)) when the second limit exists check the form again and apply the rule again

L'Hospital's rule
If <math class="calc-m"><semantics><mrow><mo movablelimits="false">lim</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim f(x) = 0</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mo movablelimits="false">lim</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim g(x) = 0</annotation></semantics></math>, or both limits are infinite, then <math class="calc-m"><semantics><mrow><mo movablelimits="false">lim</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\lim\frac{f(x)}{g(x)}</annotation></semantics></math><wbr><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mo movablelimits="false">lim</mo><mfrac><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mi>g</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">= \lim\frac{f'(x)}{g'(x)}</annotation></semantics></math>, provided the limit on the right exists (or is infinite). The derivatives of the top and bottom are taken separately.