Unit 4: Contextual Applications of Differentiation

Derivative

A measure of instantaneous rate of change of an output with respect to an input; equivalently, the slope of the tangent line to y=f(x) at a point.

Instantaneous Rate of Change

The “right now” rate at a specific input value; found using a derivative rather than an average over an interval.

Tangent Line Slope

The slope of the line that just touches a curve at a point; equals the derivative at that point, f'(a).

Average Rate of Change

Change in output over change in input on an interval: (f(b)-f(a))/(b-a).

Secant Line

The line through two points on a graph; its slope equals the average rate of change on the interval between the points.

Limit Definition of the Derivative

f'(a)=lim(h→0) [f(a+h)-f(a)]/h; the limit of secant slopes as the second point approaches the first.

Units of a Derivative

Derivative units are always (output units)/(input units), e.g., meters/second for ds/dt or dollars/item for dC/dx.

Leibniz Notation

A derivative notation that shows “with respect to what,” such as dy/dx or dV/dt; especially helpful in context problems.

Function Notation (f'(x))

A notation for the derivative of f with respect to x; means the same rate as dy/dx when y=f(x).

Second Derivative

The derivative of the derivative: f''(x)=d²y/dx²; describes how the first derivative (rate) is changing.

Positive Derivative

If f'(x)>0, the function/output is increasing as the input increases.

Negative Derivative

If f'(x)<0, the function/output is decreasing as the input increases.

Magnitude of a Derivative

How large |f'(x)| is; large magnitude means rapid change, small magnitude means slow change.

Concavity (via Second Derivative)

If f''(x)>0 then f' is increasing (often called concave up); if f''(x)<0 then f' is decreasing (often concave down).

Increasing Function

A function is increasing on an interval when f'(x)>0 on that interval.

Decreasing Function

A function is decreasing on an interval when f'(x)<0 on that interval.

Increasing at an Increasing Rate

A quantity is increasing at an increasing rate when f'(x)>0 and f''(x)>0.

Centered Difference Quotient

A table-based estimate of f'(a) using points equally spaced around a: [f(a+h)-f(a-h)]/(2h).

Net Rate (In/Out)

In contexts like tank volume, the derivative can represent the net change rate (rate in minus rate out), not necessarily one separate flow rate.

Position Function

s(t) or x(t); gives location relative to an origin (can be negative), typically measured in meters.

Velocity

v(t)=s'(t); the rate of change of position with respect to time, with units like meters/second.

Acceleration

a(t)=v'(t)=s''(t); the rate of change of velocity, with units like meters/second².

Displacement

Net change in position from t=a to t=b: s(b)-s(a).

Distance Traveled

Total ground covered regardless of direction; can be larger than |displacement| if the object turns around.

Speed

The magnitude of velocity: |v(t)|; always nonnegative.

At Rest

An object is at rest at times when its velocity is zero, v(t)=0 (not when position is zero).

Change of Direction

Occurs when velocity changes sign; times where v(t)=0 are candidates, but you must check sign on either side.

Speeding Up (1D Sign Test)

An object speeds up when velocity and acceleration have the same sign (v and a both positive or both negative).

Slowing Down (1D Sign Test)

An object slows down when velocity and acceleration have opposite signs (one positive, one negative).

Position Graph: Slope Meaning

On a position-vs-time graph, the slope at time t is velocity v(t).

Position Graph: Concavity Meaning

On a position-vs-time graph, concavity (second derivative) indicates acceleration a(t).

Velocity Graph: Height Meaning

On a velocity-vs-time graph, the y-value (height) at time t is velocity v(t).

Velocity Graph: Slope Meaning

On a velocity-vs-time graph, the slope at time t is acceleration a(t).

Chain Rule for Rates

If a variable depends on an intermediate variable, then dV/dt=(dV/dr)(dr/dt); rates must match the correct “with respect to” variable.

Intermediate Variable (in Rate Problems)

A “middle” variable (like r) that links two quantities (like V and t); mixing up dV/dr and dV/dt is a common error.

Marginal Cost

C'(q); the instantaneous rate of change of cost with respect to production level q (units: dollars per item).

Marginal Interpretation

Using a derivative as a local approximation: if C'(100)=4.2, then the 101st item costs about $4.2 more than the 100th (approximate, not exact).

Related Rates Problem

A problem where two or more quantities are linked by an equation, all change with time, and you are given some rates to find another rate at a specific instant.

Implicit Differentiation with Respect to Time

Differentiating an equation like x²+y²=25 with respect to t, producing terms like 2x(dx/dt)+2y(dy/dt)=0.

Related Rates Workflow

Read and identify givens, draw diagram, define variables, write relationship, differentiate w.r.t. time, substitute snapshot values, solve, and check units/sign.

Differentiate First, Then Substitute

A safety rule in related rates: plugging in snapshot values too early can incorrectly treat changing variables as constants.

Signed Rate Convention

“Increasing” means the rate is positive (dr/dt>0); “decreasing” means the rate is negative (dh/dt<0); signs must match the story.

Ladder Equation (Constraint)

For ladder length L with base x and height y: x²+y²=L² (L is constant).

Ladder Rate Equation

Differentiating x²+y²=L² gives 2x(dx/dt)+2y(dy/dt)=0, so dy/dt=-(x/y)(dx/dt).

Circle Area Rate Formula

If A=πr², then dA/dt=2πr(dr/dt).

Sphere Volume Rate Formula

If V=(4/3)πr³, then dV/dt=4πr²(dr/dt).

Cone Volume Formula

For a cone, V=(1/3)πr²h; in related rates, r and h may both vary and may be linked by similar triangles.

Similar Triangles Constraint (Cone)

In the example cone (height 12, radius 4), r/h=4/12=1/3, so r=h/3, letting volume be rewritten in one variable before differentiating.

Linearization (Differentials)

Using the tangent line to approximate near x: f(x+Δx)≈f(x)+f'(x)Δx (equivalently dy≈f'(x)dx).

L’Hospital’s Rule

If a limit of a quotient gives 0/0 or ∞/∞, then lim f/g = lim f'/g' (if the new limit exists), possibly applied repeatedly.