Calculus AB Homework 3.1 The Sine and Cosine Functions

Michelle Krummel

5 Nov 201722:01

EducationalLearning

32 Likes 10 Comments

TLDRThis video tutorial guides viewers through solving calculus problems involving the derivative, focusing on unit 3 homework problems 8 to 13. It explains how to find the instantaneous rate of change of an investment portfolio's value over time, using the given function V(T). The video also covers determining the concavity of the function, graphing behavior, and understanding the impact of sinusoidal terms on the function's oscillation. Additionally, it addresses finding the exact slope of a tangent line and the conditions for a function to be increasing or decreasing, using derivatives to analyze the function's behavior at specific points.

Takeaways

📈 The video covers unit 3 homework problems 8 through 13, focusing on calculus concepts.
🧮 Problem 8 involves finding the instantaneous rate of change of a portfolio's value given by V(T) = 24 * 1.07^T + 6 * sin(T) at T = 2 (two years after 2010).
🔢 The derivative V'(T) is calculated using the constant multiple rule and the exponential and trigonometric differentiation rules.
📉 V'(2) is computed to be approximately -0.638 thousand dollars per year, indicating a decrease in the portfolio's value.
🧮 The second derivative V''(T) is found to determine the concavity of the function, revealing that V''(2) is -5.33, indicating the portfolio's value is decreasing at an increasing rate.
📊 The video discusses the behavior of V(T) over the interval from 0 to 20 years and compares it to a simpler function A(T) = 24 * 1.07^T, showing that V(T) oscillates due to the sine term.
🔍 For problem 9, the exact slope of the tangent line to f(x) = 3cos(x) - 2sin(x) + 6 at x = π/4 is found by calculating the derivative.
✏️ The equation of the tangent line to y = f(x) at x = π is derived as y - 3 = 2(x - π).
📉 It's determined that f(x) is decreasing at x = π/2 because the derivative at that point is negative.
📈 The concavity of f(x) at x = 3π/2 is analyzed using the second derivative, showing that the tangent line lies above the curve as f(x) is concave down at that point.
🔢 For problem 10, derivatives of various functions involving trigonometric terms are calculated.
🔄 Problem 11 identifies values of θ where f(θ) = √3θ + 2cos(θ) has a horizontal tangent by setting the derivative equal to zero.
➕ Problems 12 and 13 use the limit definition of the derivative to identify functions and their derivatives from given limits involving trigonometric expressions.

Q & A

What is the formula given for the value of a person's investment portfolio in the video?
-The formula for the value of a person's investment portfolio is V(T) = 24 * 1.07^T + 6 * sin(T), where T represents the year and T=0 corresponds to January 1st, 2010.
What does the video aim to find for the investment portfolio on January 1st, 2012?
-The video aims to find the instantaneous rate of change of the investment portfolio's value on January 1st, 2012, which is V'(2).
How is the derivative of the exponential term in V(T) calculated in the video?
-The derivative of the exponential term 1.07^T is calculated using the exponential rule, resulting in 1.07^T * ln(1.07).
What is the result of V'(2) as calculated in the video?
-The result of V'(2) is approximately -638 dollars per year, indicating the portfolio's value is decreasing at that rate on January 1st, 2012.
What does the negative value of V'(2) signify about the portfolio's value change?
-The negative value of V'(2) signifies that the portfolio's value is decreasing at a rate of 638 dollars per year at time T equals two years, or on January 1st, 2012.
How is the second derivative V''(T) found in the video?
-V''(T) is found by differentiating V'(T), which involves differentiating each term of V'(T) using the constant multiple rule and the derivatives of sine and cosine.
What does the negative value of V''(2) indicate about the portfolio's value change over time?
-A negative value of V''(2) indicates that the portfolio's value is not only decreasing but also the rate of decrease is itself increasing, meaning the value is concave down at T equals two years.
What is the significance of the sine term in the function V(T) in the video?
-The sine term in V(T) introduces oscillations in the portfolio's value, causing it to increase and decrease on smaller intervals, unlike the strictly increasing function without the sine term.
How does the video approach finding the exact slope of the tangent line to y = f(x) at the point where x equals π/4?
-The video finds the exact slope by evaluating the derivative of f(x) at x = π/4, using the derivative of cosine and sine functions.
What does the video conclude about the function f(x) at x equals 3π/2 without graphing?
-The video concludes that the tangent line is above the curve at x equals 3π/2 by evaluating the second derivative and determining the concavity of the function at that point.
How does the video demonstrate the limit definition of the derivative for the function cosine x?
-The video demonstrates it by showing that the limit as h approaches 0 of (cosine(x+h) - cosine(x)) / h equals the derivative of cosine x, which is negative sine x.
What is the derivative of the function f(theta) = sqrt(3) * theta + 2 * cos(theta) as found in the video?
-The derivative of the function is f'(theta) = sqrt(3) - 2 * sin(theta), found by applying the constant multiple rule and the derivatives of theta and cosine theta.
What does the video conclude about the values of theta where the function f(theta) has a horizontal tangent?
-The video concludes that the function has a horizontal tangent at theta equal to π/3 and 2π/3, where the derivative equals zero.
How does the video find the limit as h approaches 0 for the function involving sine x?
-The video finds the limit by recognizing the function f(x) as 2 * sin(x) and applying the limit definition of the derivative, resulting in 2 * cos(x).

Outlines

00:00

📈 Calculating the Rate of Change in Investment Value

In this segment, the video discusses the calculation of the instantaneous rate of change of a person's investment portfolio value over time, using a given mathematical formula. The formula V(T) represents the value of the portfolio in thousands of dollars at year T, with T=0 corresponding to January 1st, 2010. The derivative V'(T) is calculated to find the rate of change at T=2 years, resulting in a negative value indicating a decrease in portfolio value. The process involves applying the constant multiple rule and exponential rule to differentiate the components of V(T). The video also explains how to interpret this negative rate of change in the context of the investment portfolio.

05:01

📉 Analyzing the Concavity and Rate of Change with Second Derivative

The video continues by determining the second derivative, V''(T), to understand how the rate of change itself is changing over time. The calculation of V''(2) reveals that the portfolio's value is not only decreasing but also doing so at an increasing rate, as indicated by a negative second derivative. This suggests the portfolio value is concave down at T=2. The explanation includes the process of differentiating V'(T) to obtain V''(T) and substituting T=2 to find the specific value, which is crucial for understanding the behavior of the investment over time.

10:07

📊 Comparing Functions and Their Impact on Portfolio Behavior

This part of the video script involves a graphical analysis of the investment function V(T) and a modified version of it, a(T), which lacks the oscillating sine component. The video describes the behavior of both functions over an interval from 0 to 20 and compares their derivatives to understand the impact of the sine term on the portfolio's value. It is shown that while V(T) oscillates between increasing and decreasing values, a(T) is strictly increasing, albeit following a similar overall trend. The derivatives' graphs further illustrate the oscillating nature of V(T) versus the consistent increase of a(T).

15:07

🔍 Derivatives and Tangent Line Approximations in Function Analysis

The script moves on to problem-solving involving the derivatives of various functions and the approximation of tangent lines. It starts with determining the exact slope of the tangent line to a function f(X) at a specific point, using the derivative evaluated at that point. The process includes substituting values into the derivative to find the slope. The video also covers finding the equation of the tangent line at another point and discusses how to determine whether a function is increasing or decreasing at a given point by evaluating the sign of the derivative. Additionally, it explores concavity and how the second derivative can indicate whether a tangent line is above or below the curve at a certain point.

20:08

🧮 Derivative Calculations and Limit Definitions

The final part of the script focuses on calculating derivatives of given functions and understanding the limit definition of the derivative. Several functions are presented, and their derivatives are computed using standard rules, such as the constant multiple rule and power rule. The video also addresses how to find the values of a variable that result in a horizontal tangent line for a function, which is when the derivative equals zero. Lastly, it demonstrates how to recognize the function from a given limit expression and correctly identify its derivative, showcasing the application of the limit definition of the derivative in various contexts.

Mindmap

Keywords

💡Derivative

The derivative is a fundamental concept in calculus that represents the rate at which a function changes at a given point. In the context of the video, the derivative is used to determine the instantaneous rate of change of the value of an investment portfolio over time. For example, when the script discusses finding 'V prime of T', it refers to the derivative of the investment value function with respect to time T, which indicates how the portfolio's value is changing at any given year T.

💡Investment Portfolio

An investment portfolio refers to a collection of financial assets such as stocks, bonds, and other securities. In the video, the value of a person's investment portfolio is represented by the function V(T), which is used to analyze how the value changes over time. The script specifically looks at the rate of change of this portfolio's value on January 1st, 2012, demonstrating the application of calculus in financial analysis.

💡Exponential Growth

Exponential growth describes a process where a quantity increases at a rate proportional to its current value. In the script, the term '1.07 to the power T' represents an exponential growth factor in the investment portfolio's value function, indicating that the portfolio's value grows multiplicatively by a factor of 1.07 each year.

💡Trigonometric Functions

Trigonometric functions, such as sine and cosine, are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the video, '6 times the sine of T' is a part of the investment portfolio's value function, introducing oscillations to the portfolio's growth, simulating fluctuations that might occur in real-world investments.

💡Instantaneous Rate of Change

The instantaneous rate of change is the rate at which a quantity changes at a specific moment in time, as opposed to its average rate of change over an interval. The video script seeks to find 'V prime of 2', which is the instantaneous rate of change of the investment portfolio's value two years after January 1st, 2010.

💡Second Derivative

The second derivative is the derivative of the first derivative of a function. It provides information about the concavity of a function and how the rate of change itself is changing. In the script, 'V double prime of T' is calculated to determine the concavity of the investment value function, which indicates whether the portfolio's value is accelerating or decelerating.

💡Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of this line is equal to the derivative of the function at that point. The video discusses finding the 'tangent line to y equals f of X' at specific points, using the derivative to determine the slope of the tangent line.

💡Concavity

Concavity refers to the curvature of a function. A function is concave up if its graph curves upward like a U, and concave down if it curves downward. The script uses the second derivative to determine the concavity of the investment portfolio's value function at a specific time, which helps in understanding the nature of the changes in the portfolio's value.

💡Limit

In calculus, a limit is the value that a function or sequence approaches as the input approaches a certain value. The script uses the concept of limits to define the derivative, as seen in problems where the limit of (f(X+h) - f(X)) / h as h approaches 0 is considered, which is the formal definition of the derivative f'(X).

💡Horizontal Tangent

A horizontal tangent to a curve is a tangent line with a slope of zero, meaning the function has a constant rate of change at that point. The video script seeks to find values of theta where the function 'f of theta' has a horizontal tangent, which occurs when the derivative of the function is equal to zero.

💡Oscillation

Oscillation in the context of functions refers to the behavior of a function that repeatedly increases and decreases. The script mentions that the term '6 times sine of T' causes the function V(T) to oscillate, indicating that the investment portfolio's value fluctuates over time, rather than following a strictly increasing or decreasing trend.

Highlights

The video discusses solving calculus problems related to investment portfolio value over time.

The formula V(t) = 24 * 1.07^t + 6 * sin(t) models the value of an investment portfolio in thousands of dollars.

The instantaneous rate of change of the portfolio's value is found using the derivative V'(t).

V'(t) is calculated using the exponential and constant multiple rules for differentiation.

The portfolio's value decreases at a rate of $638 per year on January 1st, 2012.

The second derivative, V''(t), is used to determine the concavity and rate of change of the portfolio's value.

V''(2) indicates the portfolio's value is decreasing at an increasing rate on January 1st, 2012.

The impact of the sine term in V(t) is analyzed on the portfolio's oscillating behavior over time.

Comparing the graphs of V(t) and a(t) = 24 * 1.07^t shows the effect of the oscillating sine term.

The derivative graphs of V(t) and a(t) illustrate the oscillating behavior introduced by the sine term.

The slope of the tangent line to y = f(x) at a specific point is determined using derivatives.

The tangent line approximation to y = f(x) at a point is derived using the point-slope form.

The function's increasing or decreasing nature is determined by evaluating the sign of its derivative.

The concavity of a function is assessed using the second derivative to determine the position of the tangent line relative to the curve.

Derivatives of various functions are calculated using power, constant multiple, and chain rules.

Horizontal tangents of a function are found by setting its derivative equal to zero and solving for the variable.

The limit definition of the derivative is applied to identify the derivative of cosine and 2 times sine functions.

Transcripts

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Calculus Derivative Investment Portfolio Educational Homework Rate of Change Exponential Growth Trigonometric Functions Tangent Line