Class 22

Between Class Sessions - Prep for Day 22

Please spend around 2 hours working between class sessions, focusing on the tasks below. Use any extra time to complete KnewtonAlta assignments and/or work on Project tasks.

Pick something from your prep today that you can share with your group in class. It might be something new that you learned. It might be questions you have that are still unanswered, or a question along with what helped you eventually answer it. It might be something tricky that you solved. It might be a review topic that helped you remember something. It might be a conversation you had with AI that was helpful. You will have a chance to share this with your peers during class. Come ready to articulate your thinking and questions.

Preparation

(1) Derivative Relationships (extended)

Using your sketches of the derivatives of each of the six function in this file, sketch the second derivative of each function. Remember the second derivative is the derivative of the first derivative function.

More Derivative Practice

Download this file and complete the exercises.
Consider the function \(f(y;a,b,c) = ay^2 + by + c\).
- Take the derivative with respect to \(y\), assume all other variables and parameters are constants.
- Take the derivative with respect to \(a\), assume all other variables and parameters are constants.
- Take the derivative with respect to \(c\), assume all other variables and parameters are constants.
Consider the function \(h(x,y;a) = (100 + xy)e^{-ax}\).
- Take the derivative with respect to \(x\), assume all other variables and parameters are constants.
- Take the derivative with respect to \(a\), assume all other variables and parameters are constants.
- Take the derivative with respect to \(y\), assume all other variables and parameters are constants.
Given \(f(a_1; x) = \sum_{i=1}^{3} (a_1x_i + 2)^5\), find \(\diff{f}{a_1}\) assume \(x_1\), \(x_2\), and \(x_3\) are constants.

Regular Reminders

Skill Practice (KA Homework)

Complete 2 – Linearization assignment.
If you have not done so already, complete the other derivative rule assignments
- 2 – Derivative Rules (Power Function, Sum/Difference, Constant Multiple)
- 2 – Derivative Rules (Product and Quotient)
- 2 – Derivative Rules (Function Composition) - the Chain Rule

Applied Practice (Project Work)

Review this example of a written solution for the loglikelihood calculation when using the model \(f_1\). Note this example solution is also posted in Canvas in the Additional Resources module.
Complete or Revise and then submit Project 2 Task 1
- Remember, everyone can submit Project 2 Task 1 this week, regardless of whether they have finished Project 1.
  - To earn a C, you need pass off one Project (and 2 project tasks). To earn a B, you need to pass off 2 Projects (and 4 project tasks). To earn an A, you need to pass off all 3 Project (and 6 project tasks).
- Be real with yourself.
  - If you had a rough start to the semester, other classes are a higher priority, or (fill in the blank with whatever you are dealing with this semester). Give yourself permission to focus your time and energy where you need to accomplish your goals.
  - As we move from Project 1 to Project 2 work, this is also a good time to refocus your efforts. You can move forward with Project 2, while finishing up Project 1 (Math 119 was designed to give you this personal flexibility). You can still finish all three projects this semester if you submit ready for review a Project 2 task and a Project 1 task every week and then follow-up by completing any revisions.

During Class

Derivative Rules

Brain Gains

Definition: Partial Derivative

Consider a function of several variables \(f(x,y)\).

The symbol \(\frac{\partial f}{\partial x}\) is read “the partial derivative of \(f\) with respect to \(x\)” or “the partial of \(f\) with respect to \(x\)”.
The symbol \(\frac{\partial f}{\partial y}\) is read “the partial derivative of \(f\) with respect to \(y\)” or “the partial of \(f\) with respect to \(y\)”. To calculate a partial derivative we treat all independent variables, except the one we are differentiating with respect to, as constants and use the derivative rules for a single variable function.

Let’s look at an example that includes the chain rule. Consider the function \(S(x,y) = 3(\pi x + 4y)^3\).

The partial of \(S\) with respect to \(x\) is \(\frac{\partial S}{\partial x} = 3(3(\pi x + 4y)^{2})(\pi)\).
And the partial of \(S\) with respect to \(y\) is \(\frac{\partial S}{\partial y} = 3(3(\pi x + 4y)^{2})(4)\).

For the function \(g(x) = e^{3x^2}\), identify two functions \(f(u)\) and \(u(x)\) so that \(g(x) = f(u(x))\). Then compute \(\frac{dg}{dx} = \frac{df}{du}\frac{du}{dx}\).
Use the information about \(f\) and \(f'\) in the table and \(g\) to find each of the following values.

Chain rule exercise
- Given \(h(x) = g(f(x))\), find \(h'(3)\).
- Given \(k(x) = f(g(x))\), find \(k'(1)\).

Solution

Each of these problems focus on using the chain rule. We will use \(g(1) = 2\) and \(g'(1)=3\) (slope of the line).

We have \(h'(x) = g'(f(x))f'(x)\), which means \(h'(3) = g'(f(3))f'(3) = g'(1)(6) = (3)(6) = 18\).
We get \(k'(x) = f'(g(x))g'(x)\), and so \(k'(1) = f'(g(1))g'(1) = f'(2)(3) = (5)(3) = 15\).

Given \(f(a; \textbf{x}) = \sum_{i=1}^{3} (ax_i + 2)^5\), find \(\frac{df}{da}\).

Answer

We see \(\frac{df}{da} = 5x_1(ax_1 + 2)^4 + 5x_2(ax_2 + 2)^4 + 5x_3(ax_3 + 2)^4 = \sum_{i=1}^3 5x_i(ax_i + 2)^4\).

For the function \(f(x) = e^{3x}\) and \(g(x) = x^4 - 1\), write out both \(f(g(x))\) and \(g(f(x))\), and then compute the derivative of each composite function.

Solution

The composite functions are

\(f(g(x)) = e^{3(x^4 - 1)}\) and
\(g(f(x)) = (e^{3x})^4 - 1\). The derivatives of \(f\) and \(g\) are \(f'(x) = 3e^{3x}\) and \(g'(x) = 4x^3\). The chain rule then gives us
\(\ds\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) = 3e^{3(g(x))}4x^3 = 3e^{3(x^4 - 1)}4x^3.\)
\(\ds\frac{d}{dx}[g(f(x))] = g'(f(x))f'(x) = 4(f(x))^33e^{3x} = 4(e^{3x})^33e^{3x}.\)

Group Meeting

Start by giving each person a moment to share what they chose to prepare for class. Help each other address any questions. When each person has had a chance to share, move on the other activities.

Activity - Chain Rule Practice

For each function \(g(x)\), identify two functions \(f(u)\) and \(u(x)\) so that \(g(x) = f(u(x))\) and then compute \(\frac{dg}{dx} = \frac{df}{du}\frac{du}{dx}\).

\(g(x) = (x^2-3)^4\)
\(g(x) = \ln(4x+3)\)
\(g(x) = e^{4x+3}\)
\(g(x) = (x^3+\ln x)^2\)
\(g(x) = \ln (e^{2x} + 1)\)
\(g(x) = e^{4\ln{5x}+3}\)

Activity - Partial Derivatives with the Chain Rule

Given \(S(x, y; a, b) = 3(ax + by)^{-3}\), find \(\diff{S}{x}\) assuming all other variables and parameters are constant then find \(\diff{S}{y}\) assuming all other variables and parameters are constant. (Yes, these are long-winded ways of saying compute \(\frac{\partial S}{\partial x}\) and \(\frac{\partial S}{\partial y}\).)
Given \(g(a_1, a_2; \textbf{x},\textbf{y}) = \sum_{i=1}^{7} \sqrt{a_1x_i + a_2y_i}\), find \(\frac{\partial g}{\partial a_1}\) and \(\frac{\partial g}{\partial a_2}\).
Given \(h(x, y; \textbf{a}, \textbf{b}) = \sum_{i=1}^{21} 3(a_ix + b_iy)^{-3}\), find \(\frac{\partial h}{\partial x}\) and \(\frac{\partial h}{\partial y}\).

Answer

We have \(\frac{\partial S}{\partial x}(x, y; a, b) = -9a(ax + by)^{-4}\) and \(\frac{\partial S}{\partial y}(x, y; a, b) = -9b(ax + by)^{-4}\).
We find \(\frac{\partial g}{\partial a_1} = \sum_{i=1}^7 \frac{x_i}{2\sqrt{a_1x_i + a_2y_i}}\) and \(\frac{\partial g}{\partial a_2} = \sum_{i=1}^7 \frac{y_i}{2\sqrt{a_1x_i + a_2y_i}}\).
We compute \(\frac{\partial h}{\partial x} = \sum_{i=1}^{21} -9a_i(a_ix + b_iy)^{-4}\) and \(\frac{\partial h}{\partial y} = \sum_{i=1}^{21} -9b_i(a_ix + b_iy)^{-4}\)

Activity - Graphing a Second Derivative

For the function below, graph the first and second derivative.

Solution

See Desmos.

Activity - Loglikelihood

You have been given a list of 44 data points, \((t_i, y_i)\), were \(t_i\) is the time in hours since the light was turned on and \(y_i\) is the percent intensity of the bulb after \(t_i\) hours. A model for the percent intensity of the bulb is \(f(x;m,b) = mx+b\), where \(t\) is the time in hours since the light was turned on.

Write the formula for the residuals.

Assume the residuals (errors) are independent and normally distributed (with mean of 0 and standard deviation of 1).

Write down the probability model we are using for one of the residuals.
What does knowing the residuals are independent allow us to do?
- Use the assumption of independent residuals to write the joint probability function.
Write the likelihood function for the residuals in this situation.
Write the loglikelihood function for the errors in this situation.
Rewrite the loglikelihood function using the properties of logs and sums.

Activity - Chain Rule Practice

For each pair of functions \(f(x)\) and \(g(x)\) below, write out the compositions \(f(g(x))\) and \(g(f(x))\). Then compute the derivative of each composite function. As a suggestion, compute both \(f'(x)\) and \(g'(x)\) first, and then show how you can combine \(f,g,f',g'\) to get the derivative of each composite function.
Remember to pass the chalk between each problem.

\(f(x) = 3x + 4\), \(g(x) = x^2 + 1\)
\(f(x) = e^x\), \(g(x) = \sqrt{2x-5}\)
\(f(x) = \frac{1}{x}\), \(g(x) = 2\ln(x)\)
\(f(x) = (x-5)^4\), \(g(x) = \sqrt[3]{x^2}\)
\(f(x) = e^{3x}\), \(g(x) = x^4 - 1\)
\(f(x) = \frac{x}{x-3}\), \(g(x) = \ln(5x)\)
\(f(x) = 2x^3 - 10x^2 + 2x\), \(g(x) = \sqrt{\frac{x-4}{2}}\)
\(f(x) = \frac{1}{x^5}\), \(g(x) = 4e^x\)
\(f(x) = \sqrt{8-x}\), \(g(x) = (2x-1)(x+5)\)
\(f(x) = e^{x/2}\), \(g(x) = 5x^3\)
\(f(x) = 1 - \frac{3}{x^4}\), \(g(x) = 2 - \sqrt{x}\)
\(f(x) = \frac{3}{x^2}\), \(g(x) = \ln(x)\)
\(f(x) = 27 - x^3\), \(g(x) = x^2 - x - 12\)
\(f(x) = \sqrt[3]{4-x^2}\), \(g(x) = 5x + 7\)
\(f(x) = \frac{x+1}{x-2}\), \(g(x) = e^{-x}\)

Source: Class.22 on byuimath.com