Unit 4 of 10

Applications of Differentiation

  • 12 lessons
  • Parts 65 to 88

Once you can compute derivatives, you can use them to answer real questions. Where is a function largest or smallest, when is it increasing, how fast is one quantity changing when another one does, and how well does a straight line approximate a curve.

The lessons cover extreme values, the mean value theorem, curve sketching, related rates, optimization, growth and decay models, l’Hôpital’s rule and linear approximation.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Extrema and Shape

  1. Extreme Values

    The difference between global and local maxima and minima, where on a domain they can occur, and why critical points and endpoints are the places to look.

    Theory part 65Practice part 66

  2. Extreme Value Theorems and the Closed Interval Method

    When a function is guaranteed to have a largest and a smallest value, why interior extrema sit where the derivative is zero or missing, and a short checklist that finds them on a closed interval.

    Theory part 67Practice part 68

Mean Value and Monotonicity

  1. The Mean Value Theorem

    Why a smooth function must at some point have an instantaneous rate of change equal to its average rate, which hypotheses that needs, and how it proves that a zero derivative means a constant function.

    Theory part 69Practice part 70

  2. Increasing and Decreasing Functions

    How the sign of the derivative shows where a function rises or falls, why the tests need an interval, and how a change of sign identifies local maxima and minima.

    Theory part 71Practice part 72

Extrema and Shape

  1. Concavity and Inflection Points

    How the sign of the second derivative tells you which way a graph bends, where it switches bending, and how it can classify a critical point.

    Theory part 73Practice part 74

  2. Sketching Graphs with Derivatives

    How to gather the domain, intercepts, symmetry, asymptotes, and the signs of the first and second derivatives into one honest sketch of a graph.

    Theory part 75Practice part 76

Modelling and Applications

  1. Related Rates

    How an equation linking several changing quantities turns into an equation linking their rates, and how to use it to find one rate from the others.

    Theory part 77Practice part 78

  2. Optimization Problems

    How to turn a word problem into a function on a clear domain, find its absolute maximum or minimum, and check that the answer makes sense.

    Theory part 79Practice part 80

  3. Rates of Change in the Natural and Social Sciences

    How the same derivative appears as velocity, reaction rate, growth rate, marginal cost and elasticity, and how to read its units and the assumptions behind each model.

    Theory part 81Practice part 82

  4. Exponential Growth and Decay

    Why a quantity that changes at a rate proportional to its size must be an exponential function, and how that one fact models populations, radioactive decay, cooling and interest.

    Theory part 83Practice part 84

Limits and Approximation

  1. Indeterminate Forms and l’Hôpital’s Rule

    When a limit has a form like 0 over 0 that does not decide its value, how l’Hôpital’s rule settles it, which hypotheses it needs, and how to handle products, differences and powers.

    Theory part 85Practice part 86

  2. Linear Approximations and Differentials

    How the tangent line gives a simple approximation of a function near a point, why its error shrinks faster than the step, and how to bound that error.

    Theory part 87Practice part 88