**Mathematics: Calculus and Graphical Analysis**Welcome to the calculus section of our educational website. Below you will find a detailed transcription and explanation of an advanced calculus exercise, focusing on the differentiation and graphical analysis of functions.---1. **Find $ y(5) $ and $ y'(5) $ for the given function.**---2. **Find $ x(5) $ and $ x'(5) $ for the given function.**---3. **If $ y = x^3 - 15x^2 $, find:** a) **Intervals of Increase/Decrease:** - Determine where the function is increasing and decreasing by finding the first derivative and solving for critical points. b) **Relative/Local Extrema using 1st/2nd Derivative Test:** - Locate the relative maxima and minima of the function using the first and second derivative tests. c) **Intervals of Concavity:** - Identify where the graph of the function is concave up and concave down by analyzing the second derivative. d) **Graph the function, label all points and intervals:** - Provide a precise graph of the function, including critical points, inflection points, and intervals of increase, decrease, and concavity.---4. **Differentiate the following functions:** a) $ y = e^{x}(x^3 + 4) $ d) $ f(x) = e^{-4x} $---### Explanation of Graphs and DiagramsFor item 3, the function analysis includes:- Plotting the critical points derived from the first derivative.- Using the second derivative to identify concavity changes and inflection points.- A precise graph showcasing all identified features, ensuring students can visually understand the behavior of the function.**Graphing Advice:**- Use software or graphing calculators to verify your plots.- Check your intervals for accuracy by testing points within those intervals.---The above tasks are designed to solidify your understanding of calculus concepts including differentiation, critical points, and concavity. Practice these exercises and use the graphs to visually interpret the analytical results.---

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Description: **Mathematics: Calculus and Graphical Analysis**Welcome to the calculus section of our educational website. Below you will find a detailed transcription and explanation of an advanced calculus exercise, focusing on the differentiation and graphical

Answered: 3. If y x- 15x', find; x– 15x², find:…

Math

Calculus

3. If y x- 15x', find; x– 15x², find: a) Intervals of increase/decrease.

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.4: Operations On Functions

Problem 123E

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Topic Video

Question

**Mathematics: Calculus and Graphical Analysis**

Welcome to the calculus section of our educational website. Below you will find a detailed transcription and explanation of an advanced calculus exercise, focusing on the differentiation and graphical analysis of functions.

---

1. **Find $ y(5) $ and $ y'(5) $ for the given function.**

---

2. **Find $ x(5) $ and $ x'(5) $ for the given function.**

---

3. **If $ y = x^3 - 15x^2 $, find:**

a) **Intervals of Increase/Decrease:**
- Determine where the function is increasing and decreasing by finding the first derivative and solving for critical points.

b) **Relative/Local Extrema using 1st/2nd Derivative Test:**
- Locate the relative maxima and minima of the function using the first and second derivative tests.

c) **Intervals of Concavity:**
- Identify where the graph of the function is concave up and concave down by analyzing the second derivative.

d) **Graph the function, label all points and intervals:**
- Provide a precise graph of the function, including critical points, inflection points, and intervals of increase, decrease, and concavity.

---

4. **Differentiate the following functions:**

a) $ y = e^{x}(x^3 + 4) $

d) $ f(x) = e^{-4x} $

---

### Explanation of Graphs and Diagrams

For item 3, the function analysis includes:
- Plotting the critical points derived from the first derivative.
- Using the second derivative to identify concavity changes and inflection points.
- A precise graph showcasing all identified features, ensuring students can visually understand the behavior of the function.

**Graphing Advice:**
- Use software or graphing calculators to verify your plots.
- Check your intervals for accuracy by testing points within those intervals.

---

The above tasks are designed to solidify your understanding of calculus concepts including differentiation, critical points, and concavity. Practice these exercises and use the graphs to visually interpret the analytical results.

---

Expert Solution

Step 1

We’ll answer the 3rd question since we answer only one question at a time. Please submit a new question specifying the one you’d like answered.

Given function

Step 2

Part(a)

The derivative of f(x) with respect to x.

Step 3

Part(b)

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Solved in 5 steps with 12 images

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$Angles, Arcs, and Chords and Tangents$

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