Calculus Study Guide: Maxima, Minima, Derivatives, and Optimization

Q1. Find the absolute extrema of $f(x) = x^2 + 8x$ on the interval $[-5, 3]$.

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find the maximum and minimum values of a function on a closed interval using calculus. This involves finding critical points and evaluating the function at endpoints.

Key Terms and Formulas

• Critical Point: Where $f'(x) = 0$ or $f'(x)$ is undefined.

• Absolute Extrema: The highest (maximum) and lowest (minimum) values of $f(x)$ on the interval.

• Derivative: $f'(x)$ gives the slope of the tangent line to $f(x)$.

Step-by-Step Guidance

1. Find the derivative: $f'(x) = 2x + 8$.

2. Set $f'(x) = 0$ and solve for $x$ to find critical points inside the interval.

3. Evaluate $f(x)$ at the critical point(s) and at the endpoints $x = -5$ and $x = 3$.

4. Compare these values to determine which is the largest (absolute maximum) and which is the smallest (absolute minimum).

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Q2. Find the absolute extrema of $f(x) = x^3 - 3x^2$ on the interval $[-2, 4]$.

Background

Topic: Absolute Extrema on a Closed Interval

This question is similar to Q1, but with a cubic function. You will use the same process: find critical points, evaluate at endpoints, and compare values.

Key Terms and Formulas

• Critical Point: Where $f'(x) = 0$ or $f'(x)$ is undefined.

• Absolute Extrema: The highest and lowest values of $f(x)$ on the interval.

• Derivative: $f'(x)$ gives the slope of the tangent line to $f(x)$.

Step-by-Step Guidance

1. Find the derivative: $f'(x) = 3x^2 - 6x$.

2. Set $f'(x) = 0$ and solve for $x$ to find critical points inside the interval.

3. Evaluate $f(x)$ at the critical point(s) and at the endpoints $x = -2$ and $x = 4$.

4. Compare these values to determine the absolute maximum and minimum.

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Q3. Find the intervals of increase/decrease and the x-coordinates of the local extrema for $f(x) = 2x^3 + 3x^2 - 120x + 3$.

Background

Topic: First and Second Derivative Tests

This question tests your ability to use the first derivative to find where a function is increasing or decreasing, and to use the second derivative to classify local extrema (maxima and minima).

Key Terms and Formulas

• Increasing/Decreasing: Where $f'(x) > 0$ (increasing), $f'(x) < 0$ (decreasing).

• Local Extrema: Points where $f'(x) = 0$ and the sign of $f'(x)$ changes.

• Second Derivative Test: $f''(x)$ tells you if a critical point is a max ($f''(x) < 0$) or min ($f''(x) > 0$).

Step-by-Step Guidance

1. Find the first derivative: $f'(x) = 6x^2 + 6x - 120$.

2. Set $f'(x) = 0$ and solve for $x$ to find critical points.

3. Use the second derivative $f''(x) = 12x + 6$ to classify each critical point as a local maximum or minimum.

4. Determine the intervals where $f'(x)$ is positive (increasing) or negative (decreasing).

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Q4. Find the intervals of increase/decrease and the x-coordinates of the local extrema for $f(x) = x^3 - 27x$.

Background

Topic: First and Second Derivative Tests

This question is similar to Q3, but with a different cubic function. The process is the same: use the first and second derivatives to analyze the function.

Key Terms and Formulas

• Increasing/Decreasing: Where $f'(x) > 0$ (increasing), $f'(x) < 0$ (decreasing).

• Local Extrema: Points where $f'(x) = 0$ and the sign of $f'(x)$ changes.

• Second Derivative Test: $f''(x)$ tells you if a critical point is a max ($f''(x) < 0$) or min ($f''(x) > 0$).

Step-by-Step Guidance

1. Find the first derivative: $f'(x) = 3x^2 - 27$.

2. Set $f'(x) = 0$ and solve for $x$ to find critical points.

3. Use the second derivative $f''(x) = 6x$ to classify each critical point as a local maximum or minimum.

4. Determine the intervals where $f'(x)$ is positive (increasing) or negative (decreasing).

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Q5. Find the intervals of increase/decrease and the x-coordinates of the local extrema for $f(x) = x^2 e^x$.

Background

Topic: Product Rule, First and Second Derivative Tests

This question involves differentiating a product of functions and analyzing the resulting critical points and intervals.

Key Terms and Formulas

• Product Rule: $(uv)' = u'v + uv'$

• Critical Points: Where $f'(x) = 0$ or undefined.

• Second Derivative Test: $f''(x)$ classifies extrema.

Step-by-Step Guidance

1. Find the first derivative using the product rule: $f'(x) = 2x e^x + x^2 e^x$.

2. Factor and set $f'(x) = 0$ to find critical points.

3. Use the second derivative to classify each critical point as a local maximum or minimum.

4. Determine the intervals where $f'(x)$ is positive (increasing) or negative (decreasing).

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Q6. Find the intervals of concavity and the inflection points for $f(x) = x^3 - 18x^2 + 10$.

Background

Topic: Concavity and Inflection Points

This question tests your ability to use the second derivative to determine where a function is concave up or down, and to find inflection points.

Key Terms and Formulas

• Concave Up: Where $f''(x) > 0$.

• Concave Down: Where $f''(x) < 0$.

• Inflection Point: Where $f''(x)$ changes sign.

Step-by-Step Guidance

1. Find the second derivative: $f''(x) = 6x - 36$.

2. Set $f''(x) = 0$ and solve for $x$ to find possible inflection points.

3. Test intervals around these points to determine where the function is concave up or down.

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Q7. Find the intervals of concavity and the inflection points for $f(x) = x^5 + 5x^4 - 5x + 6$.

Background

Topic: Concavity and Inflection Points

This question is similar to Q6, but with a higher-degree polynomial. The process is the same: use the second derivative to analyze concavity and inflection points.

Key Terms and Formulas

• Concave Up: Where $f''(x) > 0$.

• Concave Down: Where $f''(x) < 0$.

• Inflection Point: Where $f''(x)$ changes sign.

Step-by-Step Guidance

1. Find the second derivative: $f''(x) = 20x^3 + 60x^2$.

2. Set $f''(x) = 0$ and solve for $x$ to find possible inflection points.

3. Test intervals around these points to determine where the function is concave up or down.

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Q8. Graph and determine the intervals of increase/decrease, concavity, asymptotes, and intercepts for $f(x) = x^3 - 3x^2$.

Background

Topic: Graphing Polynomials

This question asks you to analyze a cubic function by finding its intercepts, critical points, inflection points, and intervals of increase/decrease and concavity.

Key Terms and Formulas

• Intercepts: Where the graph crosses the axes.

• Critical Points: Where $f'(x) = 0$.

• Inflection Points: Where $f''(x) = 0$.

• Intervals: Use sign charts for $f'(x)$ and $f''(x)$.

Step-by-Step Guidance

1. Find $x$- and $y$-intercepts by setting $f(x) = 0$ and $x = 0$ respectively.

2. Find critical points by solving $f'(x) = 0$.

3. Find possible inflection points by solving $f''(x) = 0$.

4. Use sign charts to determine intervals of increase/decrease and concavity.

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Q9. Graph and determine the intervals of increase/decrease, concavity, asymptotes, and intercepts for $f(x) = \frac{2x - 1}{x + 3}$.

Background

Topic: Graphing Rational Functions

This question asks you to analyze a rational function by finding its domain, intercepts, asymptotes, and intervals of increase/decrease and concavity.

Key Terms and Formulas

• Vertical Asymptote: Where the denominator is zero.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• Intercepts: Set numerator or $x = 0$.

• Critical Points: Where $f'(x) = 0$.

• Concavity: Use $f''(x)$.

Step-by-Step Guidance

1. Find the domain by setting the denominator not equal to zero.

2. Find vertical and horizontal asymptotes.

3. Find $x$- and $y$-intercepts.

4. Find critical points and intervals of increase/decrease and concavity using derivatives.

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Q10. Of all numbers whose sum is 80, find the two that have the maximum product. What is this product?

Background

Topic: Optimization

This question tests your ability to set up and solve an optimization problem using calculus. You will express the product in terms of one variable, find its maximum, and interpret the result.

Key Terms and Formulas

• Optimization: Maximizing or minimizing a quantity given a constraint.

• Product: $M = xy$ where $x + y = 80$.

• Substitute: $y = 80 - x$ so $M(x) = x(80 - x)$.

• Find maximum by setting $M'(x) = 0$.

Step-by-Step Guidance

1. Express the product $M$ in terms of one variable using the constraint.

2. Find the derivative $M'(x)$ and set it equal to zero to find the critical point.

3. Solve for $x$ and then $y$ using the constraint.

4. Calculate the maximum product using these values.

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Q11. A carpenter is building a rectangular shed with a fixed perimeter of 100 ft. What are the dimensions of the largest shed that can be built? What is its area?

Background

Topic: Optimization with Constraints

This question asks you to maximize the area of a rectangle given a fixed perimeter. This is a classic optimization problem in calculus.

Key Terms and Formulas

• Perimeter: $2x + 2y = 100$

• Area: $A = xy$

• Express $A$ in terms of one variable using the perimeter constraint.

• Find maximum by setting $A'(x) = 0$.

Step-by-Step Guidance

1. Express $y$ in terms of $x$ using the perimeter constraint.

2. Write the area $A$ as a function of $x$.

3. Find the derivative $A'(x)$ and set it equal to zero to find the critical point.

4. Solve for $x$ and $y$ to find the dimensions of the largest shed.

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Q12. A rancher has 1500 ft. of fencing to create a rectangular pen and subdivide the pen into four rectangular partitions. The pen will border a barn on one side and need no fence for that border. What dimensions of the pen will maximize the enclosed area of the pen?

Background

Topic: Optimization with Multiple Constraints

This question involves maximizing the area of a rectangular pen with a fixed amount of fencing, where the pen is subdivided and one side is bordered by a barn (so no fencing is needed on that side). This is a more advanced optimization problem.

Key Terms and Formulas

• Fencing constraint: $x + 5y = 1500$ (since there are 5 vertical partitions and 1 horizontal side to fence).

• Area: $A = xy$

• Express $A$ in terms of one variable using the constraint.

• Find maximum by setting $A'(y) = 0$.

Step-by-Step Guidance

1. Express $x$ in terms of $y$ using the fencing constraint.

2. Write the area $A$ as a function of $y$.

3. Find the derivative $A'(y)$ and set it equal to zero to find the critical point.

4. Solve for $y$ and then $x$ to find the dimensions that maximize the area.

Try solving on your own before revealing the answer!