The Central Idea
A useful curve sketch is not based on plotting many isolated points. It is built from structural information.
The main questions are:
- Where is the function defined?
- Where does it cross the axes?
- Where does it increase or decrease?
- Where does it reach local or absolute highs and lows?
- Where is it concave up or concave down?
- Where does concavity change?
- Does it have asymptotes?
- What happens as \(x\to\pm\infty\)?
Derivatives organize the local behavior, while algebra and limits organize the global behavior.
What Is an Extremum?
An extremum is a maximum or minimum value of a function.
Extrema may be:
- local;
- absolute.
The word extrema is the plural of extremum.
Local Maximum
A function \(f\) has a local maximum at \(x=c\) if \(f(c)\) is at least as large as nearby function values.
Informally, the graph reaches a local high point.
A common derivative pattern is
\[ f'(x)>0 \quad\text{to the left of }c \]and
\[ f'(x)<0 \quad\text{to the right of }c. \]The function rises into the point and falls away from it.
Local Minimum
A function \(f\) has a local minimum at \(x=c\) if \(f(c)\) is at least as small as nearby function values.
Informally, the graph reaches a local low point.
A common derivative pattern is
\[ f'(x)<0 \quad\text{to the left of }c \]and
\[ f'(x)>0 \quad\text{to the right of }c. \]The function falls into the point and rises away from it.
Absolute Maximum and Minimum
An absolute maximum is the largest function value over the entire domain or interval under consideration.
An absolute minimum is the smallest function value over that domain or interval.
Absolute extrema are global statements.
Local extrema are neighborhood statements.
Local and Absolute Extrema Can Differ
A function may have several local maxima and minima, but only one absolute highest value and one absolute lowest value, if those global extrema exist.
A local maximum does not have to be the largest value the function ever reaches.
Likewise, a local minimum does not have to be the smallest value over the full domain.
Critical Numbers Are Extremum Candidates
If \(f\) has a local extremum at an interior point \(c\) and is differentiable there, then
\[ f'(c)=0. \]More generally, local extrema can occur at critical numbers where
\[ f'(c)=0 \]or
\[ f'(c) \]does not exist.
Critical numbers are therefore candidates, not automatic extrema.
Why Candidates Must Be Classified
Consider
\[ f(x)=x^3. \]Since
\[ f'(x)=3x^2, \]we have
\[ f'(0)=0. \]But \(f\) increases through \(x=0\), so there is no local maximum or local minimum there.
A zero derivative identifies a candidate, not a conclusion.
The First-Derivative Test
The first-derivative test classifies a critical number by the sign of \(f'\) on the intervals immediately around it.
- \(+\to-\): local maximum;
- \(-\to+\): local minimum;
- \(+\to+\): no local extremum;
- \(-\to-\): no local extremum.
This test directly checks whether the function changes direction.
Example: First-Derivative Classification
Consider
\[ f(x)=x^3-3x. \]Then
\[ f'(x)=3(x-1)(x+1). \]The critical numbers are
\[ x=-1 \qquad\text{and}\qquad x=1. \]The sign pattern is
\[ +,\quad-,\quad+ \]across the intervals
\[ (-\infty,-1),\qquad(-1,1),\qquad(1,\infty). \]Classifying the Critical Points
At \(x=-1\), the derivative changes
\[ +\to-. \]Therefore, \(x=-1\) gives a local maximum.
Since
\[ f(-1)=2, \]the local maximum point is
\[ \boxed{(-1,2)}. \]At \(x=1\), the derivative changes
\[ -\to+. \]Therefore, \(x=1\) gives a local minimum.
Since
\[ f(1)=-2, \]the local minimum point is
\[ \boxed{(1,-2)}. \]The Second-Derivative Test
Suppose
\[ f'(c)=0. \]If
\[ f''(c)>0, \]then the graph is concave up at the critical point, so \(f(c)\) is a local minimum.
If
\[ f''(c)<0, \]then the graph is concave down at the critical point, so \(f(c)\) is a local maximum.
Second-Derivative Test Summary
When \(f'(c)=0\):
\[ \boxed{ f''(c)>0 \Longrightarrow \text{local minimum} } \] \[ \boxed{ f''(c)<0 \Longrightarrow \text{local maximum} }. \]If
\[ f''(c)=0, \]the test is inconclusive.
Example: Second-Derivative Test
For
\[ f(x)=x^3-3x, \]we have
\[ f''(x)=6x. \]At \(x=-1\),
\[ f''(-1)=-6<0, \]so \((-1,2)\) is a local maximum.
At \(x=1\),
\[ f''(1)=6>0, \]so \((1,-2)\) is a local minimum.
When the Second-Derivative Test Fails
Consider
\[ f(x)=x^4. \]We have
\[ f'(0)=0 \]and
\[ f''(0)=0. \]The second-derivative test gives no conclusion.
But the first derivative changes from negative to positive, so \(x=0\) is a local minimum.
Endpoint Extrema
On a closed interval, absolute extrema can occur at endpoints even though endpoints are not interior critical numbers in the ordinary derivative sense.
Therefore, absolute-extremum problems on
\[ [a,b] \]require checking:
- critical numbers inside \((a,b)\);
- the endpoint \(a\);
- the endpoint \(b\).
The Closed-Interval Method
To find absolute extrema of a continuous function on
\[ [a,b], \]- find critical numbers in \((a,b)\);
- evaluate \(f\) at each critical number;
- evaluate \(f(a)\) and \(f(b)\);
- compare all resulting values.
The largest value is the absolute maximum and the smallest is the absolute minimum.
Example: Absolute Extrema on a Closed Interval
Find the absolute extrema of
\[ f(x)=x^3-3x \]on
\[ [-2,2]. \]The critical numbers are
\[ x=-1 \qquad\text{and}\qquad x=1. \]Evaluate:
\[ f(-2)=-2, \] \[ f(-1)=2, \] \[ f(1)=-2, \] \[ f(2)=2. \]Comparing the Candidate Values
The largest function value is
\[ 2, \]attained at
\[ x=-1 \qquad\text{and}\qquad x=2. \]Therefore, the absolute maximum value is
\[ \boxed{2}. \]The smallest function value is
\[ -2, \]attained at
\[ x=-2 \qquad\text{and}\qquad x=1. \]Therefore, the absolute minimum value is
\[ \boxed{-2}. \]The Extreme Value Theorem
If a function is continuous on a closed interval
\[ [a,b], \]then it attains both an absolute maximum and an absolute minimum somewhere on that interval.
This theorem guarantees existence but does not tell us where the extrema occur.
The closed-interval method finds them.
Open Intervals May Not Have Absolute Extrema
Consider
\[ f(x)=x \]on
\[ (0,1). \]The function approaches \(0\) and \(1\), but never attains either endpoint value.
Therefore, the function has no absolute minimum and no absolute maximum on that open interval.
Curve Sketching Combines Several Layers
A reliable derivative-based sketch usually combines:
- domain;
- symmetry when useful;
- intercepts;
- critical numbers;
- increasing and decreasing intervals;
- local extrema;
- concavity;
- inflection points;
- vertical and horizontal asymptotes;
- end behavior.
Start with the Domain
Before differentiating, determine where the function exists.
Domain restrictions may come from:
- denominators equal to zero;
- even roots of negative quantities;
- logarithms of nonpositive quantities;
- piecewise definitions;
- application-specific restrictions.
The graph cannot pass through inputs outside its domain.
Intercepts Anchor the Graph
The \(y\)-intercept is found by evaluating
\[ f(0), \]when \(0\) lies in the domain.
The \(x\)-intercepts satisfy
\[ f(x)=0. \]Intercepts provide concrete points through which the graph must pass.
Symmetry Can Reduce the Work
If
\[ f(-x)=f(x), \]the function is even and its graph is symmetric about the \(y\)-axis.
If
\[ f(-x)=-f(x), \]the function is odd and its graph has origin symmetry.
Symmetry can make a curve sketch faster and easier to verify.
First Derivative: Direction
Compute
\[ f'(x). \]Then determine:
\[ f'(x)>0 \]and
\[ f'(x)<0. \]These intervals tell us where the graph rises and falls.
Second Derivative: Bending
Compute
\[ f''(x). \]Then determine:
\[ f''(x)>0 \]and
\[ f''(x)<0. \]These intervals tell us where the graph is concave up and concave down.
Inflection Points
Candidate inflection locations occur where
\[ f''(x)=0 \]or where \(f''\) is undefined.
But a point becomes an inflection point only if the concavity actually changes across it.
Vertical Asymptotes
A vertical asymptote often occurs where a function becomes unbounded near a domain break.
If
\[ \lim_{x\to a^-}f(x)=\pm\infty \]or
\[ \lim_{x\to a^+}f(x)=\pm\infty, \]then
\[ \boxed{x=a} \]is a vertical asymptote.
Horizontal Asymptotes
If
\[ \lim_{x\to\infty}f(x)=L \]or
\[ \lim_{x\to-\infty}f(x)=L, \]then
\[ \boxed{y=L} \]is a horizontal asymptote in the corresponding direction.
Slant or Oblique Asymptotes
Some rational functions approach a nonhorizontal line
\[ y=mx+b \]as \(x\to\pm\infty\).
Such a line is called a slant or oblique asymptote.
Polynomial division often reveals it directly.
End Behavior
End behavior describes what happens as
\[ x\to\infty \]and
\[ x\to-\infty. \]For polynomials, the leading term often controls the end behavior.
For rational, exponential, and logarithmic functions, limit analysis may be required.
Example: Full Curve Analysis of a Cubic
Analyze
\[ f(x)=x^3-3x. \]The domain is all real numbers.
Since
\[ f(-x)=-f(x), \]the function is odd.
Factor:
\[ f(x)=x(x^2-3). \]Therefore, the \(x\)-intercepts are
\[ \boxed{ x=-\sqrt3,\quad0,\quad\sqrt3 }. \]Direction of the Cubic
The first derivative is
\[ f'(x)=3(x-1)(x+1). \]Thus,
\[ \boxed{ f\text{ increases on } (-\infty,-1)\cup(1,\infty) } \]and
\[ \boxed{ f\text{ decreases on } (-1,1) }. \]The local maximum is
\[ \boxed{(-1,2)} \]and the local minimum is
\[ \boxed{(1,-2)}. \]Concavity of the Cubic
The second derivative is
\[ f''(x)=6x. \]Therefore,
\[ \boxed{ f\text{ is concave down on }(-\infty,0) } \]and
\[ \boxed{ f\text{ is concave up on }(0,\infty) }. \]The inflection point is
\[ \boxed{(0,0)}. \]End Behavior of the Cubic
The leading term is
\[ x^3. \]Therefore,
\[ \lim_{x\to\infty}f(x)=\infty \]and
\[ \lim_{x\to-\infty}f(x)=-\infty. \]The graph falls without bound on the left and rises without bound on the right.
Assembling the Cubic Sketch
The curve must:
- cross the \(x\)-axis at \(-\sqrt3\), \(0\), and \(\sqrt3\);
- rise to the local maximum \((-1,2)\);
- fall through the inflection point \((0,0)\);
- reach the local minimum \((1,-2)\);
- rise afterward;
- be concave down for \(x<0\);
- be concave up for \(x>0\);
- respect odd symmetry.
These constraints determine the qualitative graph far more reliably than plotting random points.
Example: Rational Curve Sketching
Consider
\[ f(x)=\frac{x}{x-1}. \]Rewrite:
\[ f(x)=1+\frac1{x-1}. \]The domain is
\[ x\neq1. \]Therefore, a vertical asymptote occurs at
\[ \boxed{x=1}. \]Since
\[ \frac1{x-1}\to0 \]as \(x\to\pm\infty\), the horizontal asymptote is
\[ \boxed{y=1}. \]Derivative Behavior of the Rational Function
Differentiate:
\[ f'(x) = -\frac1{(x-1)^2}. \]Since
\[ f'(x)<0 \]for every \(x\neq1\), the function is decreasing on both branches:
\[ \boxed{ (-\infty,1) \qquad\text{and}\qquad (1,\infty) }. \]Concavity of the Rational Function
Differentiate again:
\[ f''(x) = \frac{2}{(x-1)^3}. \]For \(x<1\),
\[ f''(x)<0, \]so the left branch is concave down.
For \(x>1\),
\[ f''(x)>0, \]so the right branch is concave up.
There is no inflection point at \(x=1\) because the function is not defined there.
Curve Sketching with Vertical Asymptotes
Near a vertical asymptote, one-sided behavior matters.
For
\[ f(x)=\frac{x}{x-1}, \]the left branch satisfies
\[ f(x)\to-\infty \qquad\text{as }x\to1^-, \]while the right branch satisfies
\[ f(x)\to\infty \qquad\text{as }x\to1^+. \]These one-sided limits determine how the branches approach the asymptote.
Example: A Quartic with Several Turning Points
Consider
\[ f(x)=x^4-4x^2. \]The function is even because
\[ f(-x)=f(x). \]Therefore, the graph is symmetric about the \(y\)-axis.
Factor:
\[ f(x)=x^2(x-2)(x+2). \]The \(x\)-intercepts are
\[ \boxed{x=-2,\quad0,\quad2}. \]Critical Points of the Quartic
Differentiate:
\[ f'(x)=4x(x^2-2). \]The critical numbers are
\[ \boxed{ x=-\sqrt2,\quad0,\quad\sqrt2 }. \]The derivative sign pattern is
\[ -,\quad+,\quad-,\quad+. \]Therefore, the graph has local minima at
\[ x=\pm\sqrt2 \]and a local maximum at
\[ x=0. \]Values of the Quartic Extrema
At
\[ x=\pm\sqrt2, \]we have
\[ f(\pm\sqrt2) = 4-8 = -4. \]So the local minima are
\[ \boxed{ (-\sqrt2,-4) \quad\text{and}\quad (\sqrt2,-4) }. \]At \(x=0\),
\[ f(0)=0. \]So the local maximum is
\[ \boxed{(0,0)}. \]Concavity of the Quartic
The second derivative is
\[ f''(x)=12x^2-8. \]The concavity changes at
\[ x=\pm\sqrt{\frac23}. \]Therefore,
\[ \boxed{ f\text{ is concave up on } \left(-\infty,-\sqrt{\frac23}\right) \cup \left(\sqrt{\frac23},\infty\right) } \]and
\[ \boxed{ f\text{ is concave down on } \left(-\sqrt{\frac23},\sqrt{\frac23}\right) }. \]End Behavior of the Quartic
The leading term is
\[ x^4. \]Therefore,
\[ \lim_{x\to\infty}f(x)=\infty \]and
\[ \lim_{x\to-\infty}f(x)=\infty. \]Both ends of the graph rise without bound.
A Sketch Is a Constraint-Satisfaction Problem
Every piece of analysis places a constraint on the graph.
The final curve must satisfy all of them simultaneously:
- domain restrictions;
- known points;
- intercepts;
- direction of motion;
- turning behavior;
- concavity;
- inflection behavior;
- asymptotic behavior;
- end behavior;
- symmetry.
Exact Coordinates Are Not Always Necessary
A qualitative sketch does not require exact coordinates for every point.
What matters is that the graph correctly represents:
- ordering of critical values;
- sign changes;
- locations of asymptotes;
- relative positions of intercepts;
- overall direction and curvature.
The sketch should be structurally correct even if it is not drawn to exact scale.
Curve Sketching from a Derivative Graph
Sometimes the graph of \(f'\) is given rather than a formula for \(f\).
Then:
- \(f'>0\) means \(f\) increases;
- \(f'<0\) means \(f\) decreases;
- zeros of \(f'\) are critical candidates;
- sign changes of \(f'\) classify turning points;
- increasing \(f'\) means \(f\) is concave up;
- decreasing \(f'\) means \(f\) is concave down.
Curve Sketching from a Sign Table
A sign table may summarize all needed information without supplying an explicit formula.
For example, suppose
\[ f': \quad +,\ -,\ + \]and
\[ f'': \quad -,\ +. \]Then the graph must:
- increase, then decrease, then increase;
- contain a local maximum at the first first-derivative sign change;
- contain a local minimum at the second;
- change concavity where the second-derivative sign changes.
Local Extrema in Applications
A local maximum or minimum may represent:
- a peak population;
- a lowest temperature;
- a maximum profit level;
- a minimum cost;
- a highest trajectory point;
- a local concentration peak;
- a minimum energy state.
The mathematics identifies the extremum; the application determines what it means.
Absolute Extrema in Optimization
Many optimization problems ask for the largest or smallest possible value of an objective function under constraints.
After the situation is modeled as a one-variable function, the same extremum tools apply:
- determine the allowed domain;
- find critical numbers;
- check boundary values;
- compare candidates.
A later optimization section will develop this modeling process in greater depth.
Common Error: Assuming Every Critical Number Is an Extremum
Critical numbers are only candidates.
The first derivative must change sign for ordinary turning behavior to occur.
The function
\[ f(x)=x^3 \]provides the standard counterexample.
Common Error: Forgetting Endpoints
On a closed interval, absolute extrema may occur at endpoints.
Checking only interior critical numbers can miss the actual global maximum or minimum.
Common Error: Using the Second-Derivative Test When It Is Inconclusive
If
\[ f''(c)=0, \]the second-derivative test gives no conclusion.
Use the first-derivative test or another argument instead.
Common Error: Calling a Discontinuity an Inflection Point
Concavity may change across a vertical asymptote, but there is no point on the graph there.
For example, \(x=1\) is not an inflection point of
\[ f(x)=\frac{x}{x-1} \]because \(f(1)\) is undefined.
Common Error: Ignoring the Domain When Sketching
A graph cannot pass through inputs excluded from the function's domain.
Domain breaks may separate the graph into distinct branches and can create asymptotic behavior.
Common Error: Drawing a Graph That Violates the Sign Chart
If
\[ f'(x)>0 \]on an interval, the graph must rise there.
If
\[ f''(x)<0, \]the graph must be concave down there.
A sketch that contradicts the derivative signs is mathematically inconsistent even if it looks visually smooth.
A Reliable Curve-Sketching Procedure
- Determine the domain.
- Check for symmetry.
- Find \(x\)- and \(y\)-intercepts.
- Identify vertical, horizontal, or slant asymptotes.
- Analyze end behavior.
- Compute \(f'(x)\).
- Find critical numbers and domain breaks.
- Build a first-derivative sign chart.
- Identify increasing and decreasing intervals.
- Classify local extrema.
- Compute \(f''(x)\).
- Build a second-derivative sign chart.
- Identify concavity and inflection points.
- Plot the important points and asymptotes.
- Draw a smooth curve consistent with every constraint.
A Reliable Procedure for Absolute Extrema on \([a,b]\)
- Verify the function is continuous on the interval when using the Extreme Value Theorem.
- Find all critical numbers inside \((a,b)\).
- Evaluate \(f\) at every interior critical number.
- Evaluate \(f(a)\) and \(f(b)\).
- Compare all values.
- Report both the extremum value and where it occurs.
Practice 1: Local Extrema
Find and classify the local extrema of
\[ f(x)=x^3-12x. \]Practice 2: Second-Derivative Test
Use the second-derivative test to classify the critical points of
\[ f(x)=x^3-3x. \]Practice 3: Absolute Extrema
Find the absolute maximum and minimum of
\[ f(x)=x^2-4x+1 \]on
\[ [0,5]. \]Practice 4: Endpoint Check
Explain why checking only critical numbers is not enough when finding absolute extrema on a closed interval.
Practice 5: Curve Features
For
\[ f(x)=x^3-3x, \]list the critical points, inflection point, increasing intervals, and decreasing interval.
Practice 6: Rational Function
For
\[ f(x)=\frac{x}{x-2}, \]identify the vertical asymptote, horizontal asymptote, and monotonicity intervals.
Practice 7: Quartic Extrema
Find the local extrema of
\[ f(x)=x^4-4x^2. \]Practice 8: Inconclusive Test
Why does the second-derivative test fail to classify \(x=0\) for
\[ f(x)=x^4? \]How can the point be classified instead?
Practice 9: Derivative Sign Information
Suppose \(f'\) has sign pattern
\[ +,\quad-,\quad+ \]across three consecutive intervals separated by critical numbers \(a\) and \(b\).
Classify the behavior at \(a\) and \(b\).
Practice 10: Identify the Error
A student finds a point where \(f''(c)=0\) and immediately labels it an inflection point.
Explain what must still be checked.
Practice Answers
Practice 1
Differentiate:
\[ f'(x)=3x^2-12=3(x-2)(x+2). \]The critical numbers are
\[ x=-2,\qquad x=2. \]The sign pattern of \(f'\) is
\[ +,\quad-,\quad+. \]Therefore, \(x=-2\) is a local maximum and \(x=2\) is a local minimum.
Evaluate:
\[ f(-2)=16, \qquad f(2)=-16. \]Thus,
\[ \boxed{\text{local maximum at }(-2,16)} \]and
\[ \boxed{\text{local minimum at }(2,-16)}. \]Practice 2
For
\[ f(x)=x^3-3x, \]the critical numbers are
\[ x=-1,\qquad x=1. \]Since
\[ f''(x)=6x, \]we have
\[ f''(-1)=-6<0, \]so \(x=-1\) gives a local maximum.
Also,
\[ f''(1)=6>0, \]so \(x=1\) gives a local minimum.
Therefore,
\[ \boxed{(-1,2)\text{ is a local maximum}} \]and
\[ \boxed{(1,-2)\text{ is a local minimum}}. \]Practice 3
Let
\[ f(x)=x^2-4x+1. \]Differentiate:
\[ f'(x)=2x-4. \]The critical number is
\[ x=2. \]Evaluate the endpoints and critical point:
\[ f(0)=1, \] \[ f(2)=-3, \] \[ f(5)=6. \]Therefore,
\[ \boxed{\text{absolute minimum }=-3\text{ at }x=2} \]and
\[ \boxed{\text{absolute maximum }=6\text{ at }x=5}. \]Practice 4
Absolute extrema may occur at endpoints even when the derivative is not used there as an interior critical-point test.
Therefore, all endpoint values must be compared with the values at interior critical numbers.
Practice 5
For
\[ f(x)=x^3-3x, \]the critical points are
\[ \boxed{(-1,2)\quad\text{and}\quad(1,-2)}. \]The inflection point is
\[ \boxed{(0,0)}. \]The function increases on
\[ \boxed{(-\infty,-1)\cup(1,\infty)} \]and decreases on
\[ \boxed{(-1,1)}. \]Practice 6
Rewrite:
\[ f(x) = 1+\frac{2}{x-2}. \]Therefore, the vertical asymptote is
\[ \boxed{x=2} \]and the horizontal asymptote is
\[ \boxed{y=1}. \]Differentiate:
\[ f'(x) = -\frac{2}{(x-2)^2}<0. \]Thus the function is decreasing on
\[ \boxed{(-\infty,2)} \]and
\[ \boxed{(2,\infty)}. \]Practice 7
For
\[ f(x)=x^4-4x^2, \]we have
\[ f'(x)=4x(x^2-2). \]The critical numbers are
\[ -\sqrt2,\qquad0,\qquad\sqrt2. \]The derivative sign pattern gives local minima at
\[ x=\pm\sqrt2 \]and a local maximum at
\[ x=0. \]Therefore,
\[ \boxed{ (-\sqrt2,-4) \text{ and } (\sqrt2,-4) \text{ are local minima} } \]and
\[ \boxed{ (0,0) \text{ is a local maximum} }. \]Practice 8
For
\[ f(x)=x^4, \]we have
\[ f'(0)=0 \]and
\[ f''(0)=0. \]Therefore, the second-derivative test is inconclusive.
The first derivative
\[ f'(x)=4x^3 \]changes from negative to positive at \(0\), so
\[ \boxed{(0,0)\text{ is a local minimum}}. \]Practice 9
At \(a\), the derivative changes
\[ +\to-, \]so \(f\) has a local maximum there.
At \(b\), the derivative changes
\[ -\to+, \]so \(f\) has a local minimum there.
Practice 10
The equation
\[ f''(c)=0 \]identifies only an inflection candidate.
The sign of \(f''\) must be checked on both sides of \(c\) to confirm that the concavity actually changes.
What to Remember
Local extrema are identified by changes in monotonicity.
The first-derivative test gives:
\[ \boxed{ +\to- \Longrightarrow \text{local maximum} } \] \[ \boxed{ -\to+ \Longrightarrow \text{local minimum} }. \]When \(f'(c)=0\), the second-derivative test gives:
\[ \boxed{ f''(c)>0 \Longrightarrow \text{local minimum} } \] \[ \boxed{ f''(c)<0 \Longrightarrow \text{local maximum} }. \]Absolute extrema on a closed interval require comparing interior critical values with endpoint values.
A complete curve sketch combines:
- domain;
- intercepts;
- symmetry;
- critical numbers;
- increasing and decreasing intervals;
- local extrema;
- concavity;
- inflection points;
- asymptotes;
- end behavior.
The graph is the visual synthesis of all this information.
Where We Go Next
Extrema and curve sketching use derivative information to describe the shape and structure of an entire function. The derivative can also be interpreted more directly as a measure of how strongly an output responds to changes in an input or parameter.
The next lesson, Sensitivity as Derivative Information, develops that interpretation across local response, parameter effects, relative sensitivity, and applied models.