I really need help figuring out how to do this problem, using the cheat sheet formulas needed for the equation **Understanding Retention of Information Based on Study Hours**The equation below represents how your retention of information varies with the number of hours you spend studying. If you want to determine when your ability to retain information (rate of retention) starts to decline, follow the instructions provided. Your task is to identify the study hour at which this decline begins. Consider what actions might be most effective after reaching this point.Given Equation: \[ M(t) = -3t^3 + 3t^2 + 12t \]This function $ M(t) $ models the relationship between study hours and retention as a cubic polynomial. The goal is to find the number of study hours $ t $ where the derivative of $ M(t) $ becomes negative, indicating a decrease in retention rate. # Calculus Cheat Sheet### The Basics- **x → f(x)**: y (y can be positive, negative, zero)- **x → f'(x)**: Slope (Slope can be positive, zero, negative) - Symbol for max or min: ![MaxMinSymbol](#)- **x → f''(x)**: Concavity (Concavity is upwards, downwards, zero) - Symbol for point of inflection: ![PointOfInflectionSymbol](#)### Derivatives- **$ y = x^n $:** $ y' = nx^{n-1} $- **$ y = u $:** $ y' = \frac{nu - n^{n-1}(u')}{u} $- **$ y = e^x $:** $ y' = e^x $- **$ y = u^e $:** $ y' = u'e^u $- **$ y = \ln u $:** $ y' = \frac{u'}{u} $### Products and Quotients- **If $ y = uv $:** $ y' = u'v + uv' $- **If $ y = \frac{u}{v} $:** $ y' = \frac{u'v - uv'}{v^2} $ - U and V are functions### Logs and Exponents- **$ y = a^u $:** $ y' = a^u \ln a $- **$ y = a^x $:** $ y' = a^x \ln a $ - Where U is a function, A is a constant, X is a variable- **$ \ln(e^x) = x $** (Simplified, not derivative)- **$ e^{\ln x} = x $** (Simplified, not derivative)- **$ \ln(MN) = \ln M + \ln N $**- **$ \ln\left(\frac{M}{N}\right) = \ln M - \ln N $** - M & N are functions (Conversions, not derivatives)- **Change of Base: \( y = \log_a x = \frac{\log x}{\log a} = \frac

we have been given the function for retaining information as a function of t -'hours of study'…

1. The following equation shows your retained information depending as a function of the hours of study. Say you wanted to find out at which point your ability to retain information (rate of retention) starts DIMINISHING. Find the number of hours of study at which this occurs. If this describes you, what is the best course of action after that M(t) = -3t³ + 3t2 + 12t amount of study?

1. The following equation shows your retained information depending as a function of the hours of study. Say you wanted to find out at which point your ability to retain information (rate of retention) starts DIMINISHING. Find the number of hours of study at which this occurs. If this describes you, what is the best course of action after that M(t) = -3t³ + 3t2 + 12t amount of study?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 43E

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Question

I really need help figuring out how to do this problem, using the cheat sheet formulas needed for the equation

**Understanding Retention of Information Based on Study Hours**

The equation below represents how your retention of information varies with the number of hours you spend studying. If you want to determine when your ability to retain information (rate of retention) starts to decline, follow the instructions provided. Your task is to identify the study hour at which this decline begins. Consider what actions might be most effective after reaching this point.

Given Equation:
\[ M(t) = -3t^3 + 3t^2 + 12t \]

This function $ M(t) $ models the relationship between study hours and retention as a cubic polynomial. The goal is to find the number of study hours $ t $ where the derivative of $ M(t) $ becomes negative, indicating a decrease in retention rate.

# Calculus Cheat Sheet

### The Basics
- **x → f(x)**: y (y can be positive, negative, zero)
- **x → f'(x)**: Slope (Slope can be positive, zero, negative)
- Symbol for max or min: ![MaxMinSymbol](#)
- **x → f''(x)**: Concavity (Concavity is upwards, downwards, zero)
- Symbol for point of inflection: ![PointOfInflectionSymbol](#)

### Derivatives
- **$ y = x^n $:** $ y' = nx^{n-1} $
- **$ y = u $:** $ y' = \frac{nu - n^{n-1}(u')}{u} $
- **$ y = e^x $:** $ y' = e^x $
- **$ y = u^e $:** $ y' = u'e^u $
- **$ y = \ln u $:** $ y' = \frac{u'}{u} $

### Products and Quotients
- **If $ y = uv $:** $ y' = u'v + uv' $
- **If $ y = \frac{u}{v} $:** $ y' = \frac{u'v - uv'}{v^2} $
- U and V are functions

### Logs and Exponents
- **$ y = a^u $:** $ y' = a^u \ln a $
- **$ y = a^x $:** $ y' = a^x \ln a $
- Where U is a function, A is a constant, X is a variable
- **$ \ln(e^x) = x $** (Simplified, not derivative)
- **$ e^{\ln x} = x $** (Simplified, not derivative)
- **$ \ln(MN) = \ln M + \ln N $**
- **$ \ln\left(\frac{M}{N}\right) = \ln M - \ln N $**
- M & N are functions (Conversions, not derivatives)
- **Change of Base: \( y = \log_a x = \frac{\log x}{\log a} = \frac

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