Processing math: 100%

Analytical Applications of Derivatives: Cheat Sheet

Download a PDF of this page here.

Download the Spanish version here.

Essential Concepts

Related Rates

  • To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.
  • In terms of the quantities, state the information given and the rate to be found.
  • Find an equation relating the quantities.
  • Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.
  • Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.

Linear Approximations and Differentials

  • A differentiable function y=f(x) can be approximated at a by the linear function
    L(x)=f(a)+f(a)(xa)
  • For a function y=f(x), if x changes from a to a+dx, then
    dy=f(x)dx

    is an approximation for the change in y. The actual change in y is

    Δy=f(a+dx)f(a)
  • A measurement error dx can lead to an error in a calculated quantity f(x). The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by
    dyf(x)dx
  • To estimate the relative error of a particular quantity q, we estimate Δqq

Maxima and Minima

  • A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
  • If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
  • A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.

The Mean Value Theorem

  • If f is continuous over [a,b] and differentiable over (a,b) and f(a)=0=f(b), then there exists a point c(a,b) such that f(c)=0. This is Rolle’s theorem.
  • If f is continuous over [a,b] and differentiable over (a,b), then there exists a point c(a,b) such that
    f(c)=f(b)f(a)ba.

    This is the Mean Value Theorem.

  • If f(x)=0 over an interval I, then f is constant over I.
  • If two differentiable functions f and g satisfy f(x)=g(x) over I, then f(x)=g(x)+C for some constant C.
  • If f(x)>0 over an interval I, then f is increasing over I. If f(x)<0 over I, then f is decreasing over I.

Derivatives and the Shape of a Graph

  • If c is a critical point of f and f(x)>0 for x<c and f(x)<0 for x>c, then f has a local maximum at c.
  • If c is a critical point of f and f(x)<0 for x<c and f(x)>0 for x>c, then f has a local minimum at c.
  • If f(x)>0 over an interval I, then f is concave up over I.
  • If f(x)<0 over an interval I, then f is concave down over I.
  • If f(c)=0 and f(c)>0, then f has a local minimum at c.
  • If f(c)=0 and f(c)<0, then f has a local maximum at c.
  • If f(c)=0 and f(c)=0, then evaluate f(x) at a test point x to the left of c and a test point x to the right of c, to determine whether f has a local extremum at c.
  • If f(x)>0 over an interval I, then f is concave up over I.
  • If f(x)<0 over an interval I, then f is concave down over I.
  • If f(c)=0 and f(c)>0, then f has a local minimum at c.
  • If f(c)=0 and f(c)<0, then f has a local maximum at c.
  • If f(c)=0 and f(c)=0, then evaluate f(x) at a test point x to the left of c and a test point x to the right of c, to determine whether f has a local extremum at c.

Key Equations

  • Linear approximation
    L(x)=f(a)+f(a)(xa)
  • A differential
    dy=f(x)dx.

Glossary

absolute extremum
if f has an absolute maximum or absolute minimum at c, we say f has an absolute extremum at c
absolute maximum
if f(c)f(x) for all x in the domain of f, we say f has an absolute maximum at c
absolute minimum
if f(c)f(x) for all x in the domain of f, we say f has an absolute minimum at c
concave down
if f is differentiable over an interval I and f is decreasing over I, then f is concave down over I
concave up
if f is differentiable over an interval I and f is increasing over I, then f is concave up over I
concavity
the upward or downward curve of the graph of a function
concavity test
suppose f is twice differentiable over an interval I; if f>0 over I, then f is concave up over I; if f<0 over I, then f is concave down over I
critical point
if f(c)=0 or f(c) is undefined, we say that c is a critical point of f
differential
the differential dx is an independent variable that can be assigned any nonzero real number; the differential dy is defined to be dy=f(x)dx
differential form
given a differentiable function y=f(x), the equation dy=f(x)dx is the differential form of the derivative of y with respect to x
extreme value theorem
if f is a continuous function over a finite, closed interval, then f has an absolute maximum and an absolute minimum
Fermat’s theorem
if f has a local extremum at c, then c is a critical point of f
first derivative test
let f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c; if f changes sign from positive to negative as x increases through c, then f has a local maximum at c; if f changes sign from negative to positive as x increases through c, then f has a local minimum at c; if f does not change sign as x increases through c, then f does not have a local extremum at c
inflection point
if f is continuous at c and f changes concavity at c, the point (c,f(c)) is an inflection point of f
linear approximation
the linear function L(x)=f(a)+f(a)(xa) is the linear approximation of f at x=a
local extremum
if f has a local maximum or local minimum at c, we say f has a local extremum at c
local maximum
if there exists an interval I such that f(c)f(x) for all xI, we say f has a local maximum at c
local minimum
if there exists an interval I such that f(c)f(x) for all xI, we say f has a local minimum at c
mean value theorem

if f is continuous over [a,b] and differentiable over (a,b), then there exists c(a,b) such that

f(c)=f(b)f(a)ba
percentage error
the relative error expressed as a percentage
propagated error
the error that results in a calculated quantity f(x) resulting from a measurement error dx
relative error
given an absolute error Δq for a particular quantity, Δqq is the relative error.
related rates
are rates of change associated with two or more related quantities that are changing over time
rolle’s theorem
if f is continuous over [a,b] and differentiable over (a,b), and if f(a)=f(b), then there exists c(a,b) such that f(c)=0
second derivative test
suppose f(c)=0 and f is continuous over an interval containing c; if f(c)>0, then f has a local minimum at c; if f(c)<0, then f has a local maximum at c; if f(c)=0, then the test is inconclusive
tangent line approximation (linearization)
since the linear approximation of f at x=a is defined using the equation of the tangent line, the linear approximation of f at x=a is also known as the tangent line approximation to f at x=a

Study Tips

Related-Rates Problem-Solving

  • Practice identifying the related quantities in word problems.
  • Memorize common geometric formulas (area, volume, Pythagorean theorem).
  • Review chain rule applications, as they’re crucial in related rates.
  • Always check if your answer makes physical sense in the context of the problem.
  • Be careful with positive/negative rates (e.g., increasing vs. decreasing quantities).

Linear Approximation of a Function at a Point

  • Visualize the tangent line and the function to understand the approximation geometrically.
  • Compare approximations with actual function values to gauge accuracy.
  • Remember that f(a) is the y-intercept and f(a) is the slope of the approximation line.
  • Use linear approximation to estimate values of complex functions or irrational numbers.

Differentials and Amount of Error

  • Practice computing differentials for various functions.
  • Understand the connection between differentials and the slope of the tangent line.
  • In error estimation problems, identify the measured quantity and the calculated quantity.
  • Remember that relative error is often more meaningful than absolute error.
  • Practice converting between absolute, relative, and percentage errors.
  • When estimating errors, consider both positive and negative variations in measurements.

Absolute Extrema

  • Practice identifying absolute extrema on various function graphs.
  • Understand the difference between the extremum (y-value) and its location (x-value).
  • Recognize scenarios where the Extreme Value Theorem applies or doesn’t apply.
  • Pay attention to the domain of the function, especially its endpoints and continuity.
  • Learn to distinguish between functions that have both, one, or no absolute extrema.

Local Extrema and Critical Points

  • Practice identifying local extrema on various function graphs.
  • Learn to distinguish between local and absolute extrema.
  • When finding critical points, check both for f(x)=0 and where f(x) is undefined.
  • Remember that critical points are only candidates for local extrema, not guarantees.
  • Pay attention to endpoints of closed intervals, as they may contain extrema but are not considered local extrema.

Locating Absolute Extrema

  • Always check both endpoints of the interval, even if they’re not critical points.
  • When finding critical points, check for both f(x)=0 and where f(x) is undefined.
  • Don’t forget to evaluate the function at each critical point within the interval.
  • Organize your work: list all candidate points (endpoints and critical points) with their function values.
  • Remember that a function can have multiple points with the same extreme value.

Rolle’s Theorem

  • Verify all conditions of Rolle’s Theorem before applying it.
  • Practice identifying functions that satisfy or violate the conditions.
  • Use Rolle’s Theorem to prove the existence of zeros of a derivative, not to find them explicitly.

The Mean Value Theorem and Its Meaning

  • Visualize the theorem geometrically: think of a secant line and parallel tangent line(s).
  • Connect the Mean Value Theorem with the concept of average rate of change.
  • Use the theorem to solve problems involving motion, particularly relating average and instantaneous velocities.
  • Remember that the theorem doesn’t tell you how to find c, just that it exists.

Corollaries of the Mean Value Theorem

  • Understand the relationship between these corollaries and the Mean Value Theorem.
  • Practice applying each corollary to different types of functions and scenarios.
  • For Corollary 1, remember that it applies to an entire interval, not just a single point.
  • For Corollary 2, think about how this relates to the concept of antiderivatives.
  • For Corollary 3, visualize how the sign of the derivative affects the graph of the function.

The First Derivative Test

  • Practice finding critical points for various function types
  • Visualize the relationship between f(x) and the graph of f(x)
  • Remember that not all critical points are local extrema
  • Use a number line to organize information about f(x) signs

Concavity and Points of Inflection

  • Practice finding f(x) for various function types
  • Visualize how f(x) relates to the “bending” of the graph
  • Remember that f(x)=0 doesn’t guarantee an inflection point
  • Use a number line to organize information about f(x) signs
  • Compare concavity analysis with first derivative analysis for a complete picture
  • Sketch graphs to confirm your analytical results

The Second Derivative Test

  • Use this test in conjunction with other analytical tools (e.g., concavity analysis)
  • Be prepared to use the First Derivative Test when this test is inconclusive