Unit 4: Contextual Applications of Differentiation

Derivative (as a rate of change)

A measurement of how one quantity changes in response to another; interpreted as the instantaneous rate of change.

Instantaneous rate of change

How fast a quantity is changing at a specific input value; equals the slope of the tangent line at that point.

Slope of the tangent line

The slope of the line that touches a curve at one point and matches the curve’s instantaneous direction there; equal to f'(a).

Limit definition of the derivative

f'(x)=lim_{h→0} [f(x+h)−f(x)]/h; defines the derivative as a limit of slopes of secant lines.

Average rate of change

The slope of the secant line on [a,b]: [f(b)−f(a)]/(b−a).

Secant line

A line through two points on a graph; its slope represents an average rate of change over an interval.

Tangent line

A line that best approximates a curve at a point; its slope represents the instantaneous rate of change.

Units of a derivative

If y has units A and x has units B, then dy/dx has units “A per B.”

Leibniz notation

Derivative written as dy/dx or df/dx; useful for tracking “what changes with respect to what.”

Prime notation

Derivative written with a prime, such as y' or f'(x), indicating the rate of change.

Operator notation

Derivative written as (d/dx)[f(x)], emphasizing differentiation as an operation.

Derivative evaluated at a point notation

The derivative at x=a can be written f'(a) or (df/dx)|_{x=a}.

Positive derivative interpretation

If f'(a)>0, then near x=a the function is increasing (the output is rising as the input increases).

Negative derivative interpretation

If f'(x)<0 on an interval, the function is decreasing there; it does NOT mean the function values are negative.

Value vs. rate confusion

f(a) is the value of the quantity at a; f'(a) is the rate at which the quantity is changing at a.

Position function

s(t) or x(t) gives an object’s location at time t (e.g., meters).

Velocity

v(t)=s'(t); the instantaneous rate of change of position (e.g., meters per second).

Acceleration

a(t)=v'(t)=s''(t); the instantaneous rate of change of velocity (e.g., meters per second squared).

Second derivative

s''(t); the derivative of the derivative, often representing acceleration when s(t) is position.

Speed

|v(t)|; the magnitude of velocity, ignoring direction.

Displacement

Net change in position over [a,b]: s(b)−s(a).

Distance traveled

Total distance along the line over [a,b]: ∫_a^b |v(t)| dt (accounts for direction changes).

Moving forward (positive direction)

An object moves forward when v(t)>0.

Moving backward (negative direction)

An object moves backward when v(t)<0.

At rest

An object is at rest when v(t)=0 (not when s(t)=0).

Speed increasing condition

Speed increases when velocity and acceleration have the same sign (v and a both positive or both negative).

Speed decreasing condition

Speed decreases when velocity and acceleration have opposite signs.

Signed area (from a velocity graph)

Area above the time-axis counts positive and below counts negative; the net signed area gives displacement.

Symmetric difference quotient

An estimate of f'(a) using nearby values on both sides: [f(a+h)−f(a−h)]/(2h).

One-sided difference quotient

An estimate of f'(a) from one side: [f(a+h)−f(a)]/h (or a left-hand version).

“Per” language in rates

Rates use “output units per input units,” e.g., joules per second for dE/dt (not seconds per joule).

Local linearity

The idea that near x=a, a differentiable function behaves almost like its tangent line.

Linearization

Using the tangent line at x=a to approximate f(x) near a.

Linear approximation formula

L(x)=f(a)+f'(a)(x−a); approximates f(x) when x is close to a.

Differentials (change approximation)

For small Δx, the change in the function is approximated by Δf ≈ f'(a)Δx.

When linear approximation can mislead

If Δx is not small or the function is highly curved near a, the tangent-line estimate can be poor.

Related rates

Problems that connect rates of change of multiple variables linked by an equation that remains true over time.

Constraint (always-true relationship)

An equation relating variables (often geometric) that holds at all times and is differentiated to link rates.

Similar triangles (related rates tool)

A common extra relationship used to reduce variables (e.g., expressing r in terms of h in a cone).

Differentiate with respect to time

In related rates, treat variables as functions of t and differentiate both sides with respect to t.

Chain rule factor in related rates

Differentiating x(t) produces terms like dx/dt; these appear because of the chain rule.

Implicit differentiation (related rates context)

Differentiating an equation with variables not solved explicitly (e.g., x^2+y^2=100) to relate dx/dt and dy/dt.

Sliding ladder relationship

For a ladder of fixed length 10 ft: x^2 + y^2 = 100, relating distance from wall and height on wall.

Area–radius rate relationship (circle)

If A=πr^2, then dA/dt = 2πr·dr/dt.

Conical tank substitution

Using similar triangles to write r=h/3, allowing V=(1/3)πr^2h to be rewritten in terms of h only.

Sphere volume rate relationship

If V=(4/3)πr^3, then dV/dt = 4πr^2·dr/dt.

Indeterminate form

A limit form like 0/0 or ∞/∞ where direct substitution does not determine the limit’s value.

L’Hospital’s Rule

For 0/0 or ∞/∞ forms, lim f/g can be found by lim f'/g' (if the new limit exists).

Conditions for using L’Hospital’s Rule

Applies to quotients where numerator and denominator both approach 0 or both approach infinity as x approaches a value.

Repeated application of L’Hospital’s Rule

If after differentiating once the limit is still 0/0 or ∞/∞, differentiate again (and re-check each time).