Increasing
\[ \boxed{ f'(x)>0 } \]The graph rises as \(x\) increases.
Calculus Guide
Applications of Derivatives
Learn how derivatives reveal the behavior of functions and solve problems involving extrema, optimization, motion, related rates, and approximation.
Putting Derivatives to Work
What Can We Do with a Derivative?
A derivative does much more than give the slope of a tangent line.
Once we know \(f'(x)\), we can determine where a function is increasing or decreasing, locate maximum and minimum values, analyze the shape of a graph, model motion, optimize quantities, and study how several changing quantities are related.
Main Idea
\[
f'(x)
\]
The sign and value of \(f'(x)\) tell us how the original function \(f(x)\) is changing.
Big Idea
Derivatives Describe Behavior
\[
f'(x)>0
\]
Increasing
\[
f'(x)<0
\]
Decreasing
\[
f'(x)=0
\]
Possible Extrema
Important Inputs
Critical Numbers
Critical numbers are the locations where important changes in a function may occur.
Critical Number
A number \(c\) in the domain of \(f\) is critical if
\[ \boxed{ f'(c)=0 \quad \text{or} \quad f'(c) \text{ does not exist} } \]Example
Find the Critical Numbers
\[ f(x)=x^3-3x^2. \]Differentiate:
\[ f'(x) = 3x^2-6x. \]Factor:
\[ f'(x) = 3x(x-2). \]Set the derivative equal to zero:
\[ 3x(x-2)=0. \] \[ x=0, \qquad x=2. \]
\[
\boxed{
x=0,\;2
}
\]
Important
Critical Does Not Automatically Mean Maximum or Minimum
A critical number is only a candidate. We still need to analyze what happens around it.
Reading the Sign of \(f'\)
Increasing and Decreasing Intervals
The sign of the first derivative tells us whether the original function rises or falls.
Decreasing
\[ \boxed{ f'(x)<0 } \]The graph falls as \(x\) increases.
Example
Analyze
\[ f(x)=x^3-3x^2. \]We already found
\[ f'(x) = 3x(x-2). \]The critical numbers divide the number line into three intervals:
\[
(-\infty,0)
\]
\[
(0,2)
\]
\[
(2,\infty)
\]
Interval
\(f'(x)\)
Behavior
\((-\infty,0)\)
\(+\)
Increasing
\((0,2)\)
\(-\)
Decreasing
\((2,\infty)\)
\(+\)
Increasing
Classifying Critical Points
The First Derivative Test
Watch how the sign of \(f'(x)\) changes as we cross a critical number.
Local Maximum
\(+\)
\(-\)
Increasing changes to decreasing.
Local Minimum
\(-\)
\(+\)
Decreasing changes to increasing.
Neither
\(+\)
\(+\)
If the sign does not change, there is no local extremum.
Continuing the Example
At \(x=0\),
\[ f'(x): \quad + \longrightarrow - \]so \(x=0\) is a local maximum.
At \(x=2\),
\[ f'(x): \quad - \longrightarrow + \]so \(x=2\) is a local minimum.
Maximum and Minimum Values
Local vs. Absolute Extrema
Local Maximum
A point whose function value is greater than nearby function values.
Local Minimum
A point whose function value is smaller than nearby function values.
Absolute Maximum
The largest function value on the entire interval or domain being considered.
Absolute Minimum
The smallest function value on the entire interval or domain being considered.
Absolute Extrema
The Closed Interval Method
To find absolute extrema on a closed interval \([a,b]\), compare the function values at every critical number and at both endpoints.
1
Differentiate
Find \(f'(x)\)
2
Find Candidates
Locate Critical Numbers
Keep only critical numbers inside the interval.
3
Evaluate
Plug Candidates into \(f\)
Include both endpoints.
4
Compare
Largest and Smallest Values Win
Example
Find the Absolute Extrema of \(f(x)=x^3-3x^2\) on \([-1,3]\)
Critical numbers:
\[ x=0,\;2. \]Evaluate at the endpoints and critical numbers:
\[ f(-1)=-4 \] \[ f(0)=0 \] \[ f(2)=-4 \] \[ f(3)=0. \]Absolute Maximum
\[ \boxed{0} \]at \(x=0\) and \(x=3\)
Absolute Minimum
\[ \boxed{-4} \]at \(x=-1\) and \(x=2\)
The Second Derivative
Concavity
The second derivative describes how the slope of a function is changing.
Concave Up
\[ \boxed{ f''(x)>0 } \]Slopes are increasing.
Concave Down
\[ \boxed{ f''(x)<0 } \]Slopes are decreasing.
Keep Them Separate
The first derivative tells us whether \(f\) is increasing or decreasing. The second derivative tells us whether the graph is concave up or concave down.
Changing Concavity
Inflection Points
Inflection Point
A point where the graph changes concavity.
Example
Analyze
\[ f(x)=x^3. \]First derivative:
\[ f'(x)=3x^2. \]Second derivative:
\[ f''(x)=6x. \]Set \(f''(x)=0\):
\[ 6x=0 \] \[ x=0. \]For \(x<0\),
\[ f''(x)<0, \]while for \(x>0\),
\[ f''(x)>0. \]The concavity changes, so \((0,0)\) is an inflection point.
\[
\boxed{
(0,0)
}
\]
Common Mistake
\(f''(c)=0\) Is Not Enough
You must verify that the concavity actually changes across \(x=c\).
Another Extrema Test
The Second Derivative Test
If \(f'(c)=0\), the second derivative can sometimes classify the critical point quickly.
Build the Graph from Calculus
Curve Sketching with \(f'\) and \(f''\)
Derivatives allow us to predict the important shape of a graph even before plotting many points.
1
Domain and Intercepts
Start with the basic algebraic features of \(f\).
2
Critical Numbers
Find where \(f'=0\) or \(f'\) does not exist.
3
Increasing / Decreasing
Build a sign chart for \(f'\).
4
Local Extrema
Use sign changes or the second derivative test.
5
Concavity
Analyze the sign of \(f''\).
6
Inflection Points
Locate actual changes in concavity.
Maximize or Minimize
Optimization
Optimization problems ask us to find the largest or smallest possible value of a quantity.
Optimization Strategy
Build One Function in One Variable
- Identify the quantity to maximize or minimize.
- Write an objective function.
- Use the given constraint to rewrite the objective function using one variable.
- Determine the meaningful domain.
- Differentiate.
- Find critical numbers.
- Determine which candidate gives the desired maximum or minimum.
- Answer the original question with appropriate units.
Example
Maximum Area with a Fixed Perimeter
A rectangle has perimeter \(40\) units. Find the dimensions that maximize its area.
Let length be \(x\) and width be \(y\).
\[ 2x+2y=40. \]Solve for \(y\):
\[ y=20-x. \]Area:
\[ A=xy. \] \[ A(x) = x(20-x). \] \[ A(x) = 20x-x^2. \]Differentiate:
\[ A'(x) = 20-2x. \]Set equal to zero:
\[ 20-2x=0. \] \[ x=10. \]Then
\[ y=20-10=10. \]
\[
\boxed{
10
\text{ units}
\times
10
\text{ units}
}
\]
Position, Velocity, Acceleration
Motion Along a Line
Position
\[ s(t) \]Where the object is.
Velocity
\[ v(t) = s'(t) \]How position changes.
Acceleration
\[ a(t) = v'(t) = s''(t) \]How velocity changes.
Moving Right
\[ v(t)>0 \]Moving Left
\[ v(t)<0 \]At Rest
\[ v(t)=0 \]Speeding Up
\(v\) and \(a\) have the same sign.
Slowing Down
\(v\) and \(a\) have opposite signs.
Example
Suppose
\[ s(t) = t^3-6t^2+9t. \]Velocity:
\[ v(t) = 3t^2-12t+9. \]Acceleration:
\[ a(t) = 6t-12. \]To find when the object is at rest, solve
\[ v(t)=0. \] \[ 3t^2-12t+9=0. \] \[ 3(t-1)(t-3)=0. \]
\[
\boxed{
t=1,\;3
}
\]
Tangent Lines as Approximations
Linear Approximation
Near a point where a function is differentiable, the tangent line can approximate nearby function values.
Linearization at \(x=a\)
\[ \boxed{ L(x) = f(a) + f'(a)(x-a) } \]Example
Approximate \(\sqrt{4.1}\)
Let
\[ f(x)=\sqrt{x}. \]Use \(a=4\), because \(\sqrt{4}=2\).
\[ f(4)=2. \]Differentiate:
\[ f'(x) = \frac{1}{2\sqrt{x}}. \] \[ f'(4) = \frac14. \]Build the linearization:
\[ L(x) = 2 + \frac14(x-4). \]Evaluate at \(x=4.1\):
\[ L(4.1) = 2 + \frac14(0.1). \] \[ L(4.1)=2.025. \]
\[
\boxed{
\sqrt{4.1}
\approx
2.025
}
\]
Watch Out
Common Mistakes
1
Treating Every Critical Number as an Extremum
A critical number is only a candidate. Check the behavior around it.
2
Forgetting Where \(f'\) Is Undefined
Critical numbers can occur where \(f'(x)=0\) or where \(f'\) does not exist, as long as \(f\) itself is defined.
3
Using \(f'\) for Concavity
Increasing and decreasing use \(f'\). Concavity uses \(f''\).
4
Calling Every \(f''=0\) Point an Inflection Point
Concavity must actually change.
5
Forgetting Endpoints
Absolute extrema on a closed interval require checking both endpoints.
6
Optimizing the Wrong Quantity
Clearly identify the objective function before differentiating.
7
Substituting Too Early in Related Rates
Differentiate the relationship first, then substitute the values for the instant being considered.
8
Forgetting Units
Optimization, motion, and related-rate answers should include meaningful units.
Problem-Solving Roadmap
Applications of Derivatives Strategy
1
Identify
What Is the Problem Asking About?
Extrema, intervals, concavity, optimization, motion, related rates, or approximation?
2
Differentiate
Decide Whether You Need \(f'\) or \(f''\)
3
Find Important Values
Solve for Candidates
Critical numbers, possible inflection points, endpoints, or meaningful times.
4
Analyze
Use Signs or Compare Values
5
Interpret
Answer the Original Question
State intervals, points, dimensions, rates, or units clearly.
Keep This Handy
Applications Quick Reference
Increasing
\[ f'(x)>0 \]Decreasing
\[ f'(x)<0 \]Critical Number
\[ f'(c)=0 \]or \(f'(c)\) does not exist
Local Maximum
\[ f': + \to - \]Local Minimum
\[ f': - \to + \]Concave Up
\[ f''(x)>0 \]Concave Down
\[ f''(x)<0 \]Velocity
\[ v(t)=s'(t) \]Acceleration
\[ a(t)=s''(t) \]Linear Approximation
\[ L(x) = f(a) + f'(a)(x-a) \]Before You Practice
Applications of Derivatives Checklist
- Find critical numbers from \(f'(x)=0\) or undefined derivatives.
- Use the sign of \(f'\) to determine increasing and decreasing intervals.
- Use the First Derivative Test to classify local extrema.
- Distinguish local extrema from absolute extrema.
- Use the Closed Interval Method to find absolute extrema.
- Use \(f''\) to determine concavity.
- Verify actual changes in concavity before identifying inflection points.
- Use the Second Derivative Test when its conditions apply.
- Combine \(f'\) and \(f''\) to analyze and sketch curves.
- Build optimization problems using one objective function and one variable.
- Interpret position, velocity, and acceleration correctly.
- Differentiate related-rate equations with respect to time.
- Use tangent lines for linear approximation.
- Include units and interpret the final result in context.
Your Turn
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Practice critical points, extrema, concavity, optimization, motion, related rates, and linear approximation.
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