• Graph slices (holding one var constant) is important for understanding partial derivatives
• Contour plots

• Vector Field: basically a way of visualizing functions that have the same number of inputs as outputs (outputs are vectors).

• Partial Derivative:
  • Interpret as: how does a tiny change in the input in the \(x\)OR\(y\) direction influence the output \(f\); keeping the other variable constant. (technically, you then take the ratio of the nudges)
  • On a graph, interpret as slicing the graph at the constant variable and then looking at the slope on the projected/sliced graph.
  • Properties:
    • \[f_{x y} = f_{y x}\]
• Tangent Hyperplanes:
  The tangent hyperplane to a curve at a given point \(\mathbf{x}\) is the best linear approximation of the curve at that point.
  • Tangent Line:

    $$y-y_{0}=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)$$

  • Tangent Plane:

    $$y-y_{0}=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)$$

    of the surface \(z=f(x, y)\) at the point \(P\left(x_{0}, y_{0}, z_{0}\right)\)

• Gradients:
  • Properties:
    • Always normal to contour lines

FIRST

1. Asynchronous:

2. Asynchronous:

3. Asynchronous:

4. Asynchronous:

5. Asynchronous:

6. Asynchronous:

7. Asynchronous:

8. Asynchronous:

SECOND

1. Asynchronous:

2. Asynchronous:

3. Asynchronous:

4. Asynchronous:

5. Asynchronous:

6. Asynchronous:

7. Asynchronous:

8. Asynchronous:

THIRD

1. Asynchronous:

2. Asynchronous:

3. Asynchronous:

4. Asynchronous:

5. Asynchronous:

6. Asynchronous:

7. Asynchronous:

8. Asynchronous: