Applications of Differentiation: Rates, Optimization, Curve Sketching, and Approximations

1. Introduction to Applications of Differentiation

Differentiation is not just about finding derivatives—it is a powerful tool for modeling and solving real-world problems. This unit covers four key applications:

1. Related Rates: How quantities change with respect to time.
2. Optimization: Finding maxima/minima in practical scenarios.
3. Curve Sketching: Using derivatives to analyze and sketch graphs.
4. Linear Approximations: Approximating functions near a point.

These applications rely on understanding derivatives, critical points, and concavity, which were introduced in Unit 3 (Differentiation).

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2. Related Rates

2.1 Definition and Concept

Related rates problems involve finding the rate of change of one quantity when another related quantity changes. The key steps are:

1. Identify the given rate (usually involving time, $t$).
2. Express all quantities in terms of one variable.
3. Differentiate both sides with respect to time ($t$) using the chain rule.
4. Solve for the unknown rate.

Example Scenario: A spherical balloon is being inflated. If the radius increases at $2\text{cm/s}$, how fast is the volume increasing when the radius is $5\text{cm}$?

2.2 Worked Example: Balloon Volume Problem

Given:

• Radius $r=5\text{cm}$
• $\frac{dr}{dt}=2\text{cm/s}$
• Volume of a sphere: $V=\frac{4}{3}\pi r^3$

Solution:

1. Differentiate $V$ with respect to $t$: $$\frac{dV}{dt}=\frac{d}{dt}\left(\frac{4}{3}\pi r^3\right)=4\pi r^2\frac{dr}{dt}$$
2. Substitute $r=5$ and $\frac{dr}{dt}=2$: $$\frac{dV}{dt}=4\pi (5{)}^2(2)=200\pi {\text{cm}}^3/\text{s}$$

Answer: The volume increases at $200\pi {\text{cm}}^3/\text{s}$.

2.3 Common Pitfalls

• Forgetting to express everything in terms of one variable (e.g., mixing $r$ and $V$ without relating them).
• Misapplying the chain rule (e.g., differentiating $r^3$ as $3r^2$ without multiplying by $\frac{dr}{dt}$).
• Ignoring units in the final answer.

2.4 Applications of Related Rates

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3. Optimization Problems

3.1 Definition and Approach

Optimization involves finding the maximum or minimum value of a function in a given context. Steps:

1. Define the objective function (e.g., profit, area, cost).
2. Express it in terms of a single variable using constraints.
3. Find critical points by setting the derivative to zero ($f^{\prime }(x)=0$).
4. Use the second derivative test or analyze intervals to determine maxima/minima.
5. Check endpoints if the domain is closed.

3.2 Worked Example: Minimizing Cost

Problem: A farmer wants to fence a rectangular field with 100 meters of fencing. What dimensions maximize the area?

Solution:

1. Let length = $L$, width = $W$.
2. Constraint: $2L+2W=100$ → $W=50-L$.
3. Area $A=L\times W=L(50-L)=50L-L^2$.
4. Find critical points: $$\frac{dA}{dL}=50-2L=0⟹L=25$$ Then $W=25$.
5. Second derivative: $\frac{d^{2}A}{d{L}^2}=-2<0$ → maximum area.

Answer: A square field ($25\text{m}\times 25\text{m}$) maximizes the area.

3.3 Comparison: Absolute vs. Local Extrema

3.4 Applications of Optimization

• Business: Maximizing profit, minimizing cost.
• Engineering: Optimal design (e.g., strongest beam with least material).
• Biology: Maximizing population growth under constraints.
• Physics: Least time path (Brachistochrone problem).

Example Problem: A box with no top is to be made from a square piece of cardboard with side length $10\text{cm}$. Squares of side $x$ are cut from each corner, and the sides are folded up. What $x$ maximizes the volume?

Solution:

1. Volume $V=x(10-2x{)}^2$.
2. $\frac{dV}{dx}=(10-2x{)}^2+x\cdot 2(10-2x)(-2)=0$.
3. Solve $100-40x+4x^2-4x(10-2x)=0$ → $x=2.5\text{cm}$.

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4. Curve Sketching Using Derivatives

4.1 Steps to Sketch a Graph

To graph $f(x)$, use:

1. First derivative ($f^{\prime }(x)$): Sign tells increasing/decreasing; zeros give critical points.
2. Second derivative ($f^{\prime \prime }(x)$): Concavity (up/down); zeros give inflection points.
3. Intercepts: Solve $f(x)=0$ (x-intercepts) and $x=0$ (y-intercept).
4. Asymptotes: Horizontal (${\lim }_{x\to \pm \infty }f(x)$), vertical (${\lim }_{x\to a}f(x)=\infty$).
5. Behavior at critical points: Use the first derivative test or second derivative test.

4.2 Worked Example: Sketching $f(x)=x^3-3x^2$

1. Find $f^{\prime }(x)$ and $f^{\prime \prime }(x)$: $$f^{\prime }(x)=3x^2-6x,f^{\prime \prime }(x)=6x-6$$
2. Critical points: $f^{\prime }(x)=0$ → $x=0,2$.
3. Inflection point: $f^{\prime \prime }(x)=0$ → $x=1$.
4. Sign analysis:
   • $f^{\prime }(x)>0$ for $x<0$ and $x>2$ → increasing.
   • $f^{\prime }(x)<0$ for $0<x<2$ → decreasing.
   • $f^{\prime \prime }(x)>0$ for $x>1$ → concave up.
   • $f^{\prime \prime }(x)<0$ for $x<1$ → concave down.
5. Key points:
   • $x=0$: Local max ($f(0)=0$).
   • $x=2$: Local min ($f(2)=-4$).
   • $x=1$: Inflection point ($f(1)=-2$).
6. Intercepts: $f(x)=0$ → $x=0,3$.

Sketch:

y
|
4 |       /
3 |      /
2 |     /
1 |    /
0 |---+---+---+---+--- x
  -1  0   1   2   3
     |   |   |   |
     |   |   I   |
     |   |   P   |
     |   L   M   |
     |   o     o
     |   max   min

4.3 Common Mistakes

• Ignoring asymptotes: Always check limits at infinity and vertical asymptotes.
• Misclassifying critical points: Use the first derivative test if $f^{\prime \prime }(x)=0$ (e.g., $f(x)=x^4$ at $x=0$).
• Forgetting to plot intercepts: Always find where $f(x)=0$ and $x=0$.

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5. Linear Approximations (Tangent Line Approximation)

5.1 Definition

The linear approximation (or tangent line approximation) of $f(x)$ near $x=a$ is: $$L(x)=f(a)+f^{\prime }(a)(x-a)$$ This is the first-order Taylor polynomial and is useful for estimating $f(x)$ when $x$ is close to $a$.

5.2 Worked Example: Approximating $\sqrt{16.1}$

Problem: Approximate $\sqrt{16.1}$ using $a=16$.

Solution:

1. Let $f(x)=\sqrt{x}$, $a=16$.
2. $f(16)=4$.
3. $f^{\prime }(x)=\frac{1}{2\sqrt{x}}$ → $f^{\prime }(16)=\frac{1}{8}$.
4. Linear approximation: $$L(x)=4+\frac{1}{8}(x-16)$$
5. For $x=16.1$: $$L(16.1)=4+\frac{1}{8}(0.1)=4.0125$$ Actual value: $\sqrt{16.1}\approx 4.0125$ (very close!).

5.3 Error in Approximation

The error $E(x)$ is given by: $$E(x)=f(x)-L(x)\approx \frac{f^{\prime \prime }(a)}{2}(x-a{)}^2$$ For $f(x)=\sqrt{x}$, $f^{\prime \prime }(x)=-\frac{1}{4}{x}^{-3/2}$, so: $$E(16.1)\approx \frac{-1/(4\cdot 64)}{2}(0.1{)}^2=-2.08\times 10^{-5}$$ The approximation is very accurate near $a$.

5.4 Applications

• Physics: Approximating small changes in systems.
• Economics: Estimating marginal cost/revenue.
• Engineering: Small-angle approximations (e.g., $\sin \theta \approx \theta$ for $\theta$ in radians).

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6. Summary Table of Applications

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7. Exam Tips for Unit 4

7.1 Common Exam Questions

1. Related Rates:

   • Expect problems involving geometric shapes (spheres, cones, cylinders) or physics scenarios (ladder sliding, water filling).
   • Always draw a diagram if geometry is involved.
   • Units matter: Ensure your final answer has consistent units (e.g., ${\text{cm}}^3/\text{s}$).

2. Optimization:

   • Read carefully: Identify the objective (max/min) and constraints.
   • Check endpoints: If the domain is closed (e.g., $[a,b]$), evaluate $f$ at critical points and endpoints.
   • Economic problems: Often involve profit = revenue - cost.

3. Curve Sketching:

   • Show all steps: List $f^{\prime }(x)$, $f^{\prime \prime }(x)$, critical points, and concavity.
   • Label everything: Mark intercepts, asymptotes, and critical points on the sketch.
   • Use test points: For $f^{\prime }(x)$ and $f^{\prime \prime }(x)$, pick values in intervals to determine signs.

4. Linear Approximations:

   • Memorize the formula: $L(x)=f(a)+f^{\prime }(a)(x-a)$.
   • Small changes: The approximation is best when $|x-a|$ is small.
   • Error analysis: If asked, compute $f^{\prime \prime }(a)$ for the error term.

7.2 High-Scoring Strategies

• For related rates:
  • Write the relationship between variables before differentiating.
  • Box your final answer with units.
• For optimization:
  • Clearly state whether you’re finding a max or min.
  • Verify with the second derivative test if possible.
• For curve sketching:
  • Organize your work in a table:

    $x$	$f(x)$	$f^{\prime }(x)$	$f^{\prime \prime }(x)$	Behavior
    ...	...	...	...	Increasing

  • Sketch lightly first, then darken final lines.
• For linear approximations:
  • Show the tangent line equation explicitly.
  • Compare with exact value if possible to demonstrate accuracy.

7.3 Pitfalls to Avoid

• Assuming all critical points are maxima/minima: Use the first or second derivative test.
• Forgetting to check endpoints in optimization problems.
• Skipping units in related rates problems.
• Overcomplicating curve sketches: Focus on key features (intercepts, critical points, asymptotes).
• Misapplying the chain rule in related rates (e.g., forgetting $\frac{dr}{dt}$).

7.4 Practice Problems for Revision

1. Related Rates:
   • A ladder 10 m long leans against a wall. If the bottom slides away at $1\text{m/s}$, how fast is the top sliding down when the bottom is 6 m from the wall?
2. Optimization:
   • Find the dimensions of a rectangular box with volume $1000{\text{cm}}^3$ that minimize the surface area.
3. Curve Sketching:
   • Sketch $f(x)=x^4-4x^3$. Include intercepts, critical points, and inflection points.
4. Linear Approximation:
   • Approximate $\sin (0.1)$ using $a=0$.

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8. References and Further Reading

• Textbooks:
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage.
  • Thomas, G. B., Weir, M. D., & Hass, J. (2014). Thomas' Calculus. Pearson.
• Online Resources:
  • Khan Academy: Applications of Derivatives
  • Paul’s Online Math Notes: Optimization
• Nepali Context:
  • Tribhuvan University past question papers (especially MTH117).
  • PU notes on applied calculus for real-world problem-solving.

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9. Quick Revision Checklist

Before the exam, ensure you can: ✅ Differentiate implicitly and apply the chain rule in related rates. ✅ Set up and solve optimization problems with constraints. ✅ Sketch a graph given $f(x)$, $f^{\prime }(x)$, and $f^{\prime \prime }(x)$. ✅ Write the linear approximation formula and compute errors. ✅ Recognize when to use the first vs. second derivative test.

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Final Note: Applications of differentiation are problem-solving oriented. Practice mixing concepts (e.g., optimization + related rates) to build confidence. Good luck! 🚀