Calculus Chapter 3 Review: Critical Values, Extrema, Concavity, and Applications

Q1. Sketch the graph that possesses the characteristics listed:

It is concave down at (1,4), concave up at (5,-6), and has an inflection point at (3,-1). Choose the correct graph below.

Background

Topic: Curve Sketching & Concavity

This question tests your understanding of how to interpret and sketch a graph based on information about concavity and inflection points.

Key Terms and Formulas:

• Concave Up: The graph is shaped like a cup (holds water), $f''(x) > 0$.

• Concave Down: The graph is shaped like a cap (spills water), $f''(x) < 0$.

• Inflection Point: The point where the graph changes concavity, $f''(x) = 0$ and changes sign.

Step-by-Step Guidance

1. Mark the given points on a coordinate plane: (1,4), (5,-6), and (3,-1).

2. At (1,4), the graph should be concave down. At (5,-6), it should be concave up. At (3,-1), the graph should change concavity (inflection point).

3. Sketch a curve that is concave down to the left of (3,-1), passes through (1,4), changes concavity at (3,-1), and is concave up to the right, passing through (5,-6).

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Q2. Find a) any critical values and b) any relative extrema for $y = x^4 - 11x^2$

Background

Topic: Critical Points & Relative Extrema

This question tests your ability to find critical values (where $f'(x) = 0$ or $f'(x)$ is undefined) and determine relative maxima and minima using calculus.

Key Terms and Formulas:

• Critical Value: $x$ where $f'(x) = 0$ or $f'(x)$ does not exist.

• Relative Extrema: Local maximum or minimum points, found by analyzing the sign of $f'(x)$ or using the second derivative test.

• First Derivative: $f'(x)$ gives the slope of the tangent line and helps find critical points.

Step-by-Step Guidance

1. Find the first derivative: $f'(x) = \frac{d}{dx}(x^4 - 11x^2)$.

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

3. Use the second derivative or sign chart to determine if each critical value is a relative maximum, minimum, or neither.

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Q3. For the function $y = x^4 - 11x^2$, analyze the graph:

• a) Find the coordinates of the relative extrema.

• b) Identify intervals where the function is increasing or decreasing.

• c) Find the coordinates of the inflection points.

• d) Identify intervals where the function is concave up or concave down.

• e) Choose the correct graph.

Background

Topic: Curve Sketching, Extrema, and Concavity

This question tests your ability to use derivatives to analyze and sketch the graph of a function, including finding extrema, inflection points, and intervals of increase/decrease and concavity.

Key Terms and Formulas:

• First Derivative Test: Determines intervals of increase/decrease and relative extrema.

• Second Derivative Test: Determines concavity and inflection points.

Step-by-Step Guidance

1. Find $f'(x)$ and solve $f'(x) = 0$ for critical points.

2. Test intervals around critical points to determine where the function is increasing or decreasing.

3. Find $f''(x)$ and solve $f''(x) = 0$ for possible inflection points.

4. Test intervals around inflection points to determine concavity (up or down).

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Q4. Find the absolute maximum and minimum values of $f(x) = 15x^4 - 4x^3$ on $[-2,2]$

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find the highest and lowest values of a function on a closed interval using calculus.

Key Terms and Formulas:

• Absolute Maximum/Minimum: The largest/smallest value of $f(x)$ on the interval.

• Closed Interval Method: Evaluate $f(x)$ at critical points inside the interval and at the endpoints.

Step-by-Step Guidance

1. Find $f'(x)$ and solve $f'(x) = 0$ for critical points in $[-2,2]$.

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

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

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Q5. For the function $h(x) = x e^{5x}$, find any relative extrema.

Background

Topic: Relative Extrema of Exponential Functions

This question tests your ability to use derivatives to find and classify relative extrema for a function involving an exponential term.

Key Terms and Formulas:

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Critical Value: $x$ where $h'(x) = 0$ or $h'(x)$ does not exist.

Step-by-Step Guidance

1. Find $h'(x)$ using the product rule.

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

3. Use the second derivative or sign chart to classify each critical value as a relative maximum or minimum.

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Q6. An employee's monthly productivity $M(t)$ is given by $M(t) = -3t^2 + 150t + 120$, $0 0  t  40$. Find the maximum productivity and the year in which it is achieved.

Background

Topic: Optimization (Quadratic Functions)

This question tests your ability to find the maximum value of a quadratic function, which models productivity over time.

Key Terms and Formulas:

• Vertex of a Parabola: For $M(t) = at^2 + bt + c$, the maximum (if $a < 0$) occurs at $t = -\frac{b}{2a}$.

Step-by-Step Guidance

1. Identify the coefficients $a$, $b$, and $c$ in the quadratic function.

2. Use the vertex formula $t = -\frac{b}{2a}$ to find the year when maximum productivity occurs.

3. Plug this value of $t$ back into $M(t)$ to find the maximum productivity.

Try solving on your own before revealing the answer!