Week 7 Checkpoint

The chain rule, computational graphs with forward/backward passes on $y=(wx+b)^2$, partial derivatives, the gradient vector, the direction of steepest ascent, and reading contour plots.

  1. 1.

    Use the chain rule to differentiate y=sin⁡(x2)y=\sin(x^2).

  2. 2.

    For y=(wx+b)2y=(wx+b)^2 with w=2, x=3, b=1w=2,\ x=3,\ b=1, the forward pass gives u=wx+b=7u=wx+b=7 and y=49y=49. Using the chain rule on the backward pass, what is ∂y∂w\frac{\partial y}{\partial w}?

  3. 3.

    For f(x,y)=x2y+3yf(x,y)=x^2 y + 3y, what is the partial derivative ∂f∂x\frac{\partial f}{\partial x}?

  4. 4.

    The gradient ∇f\nabla f of f(x,y)=x2+y2f(x,y)=x^2+y^2 evaluated at the point (1,2)(1,2) is:

  5. 5.

    At a point where ∇f≠0\nabla f\neq 0, the direction of steepest ascent of ff is:

  6. 6.

    On a contour (level-curve) plot of ff, the gradient vector at a point is always:

  7. 7.

    In reverse-mode autodiff, the backward pass multiplies the upstream gradient by each node's local derivative. For the node y=u2y=u^2 with upstream gradient ∂L∂y=g\frac{\partial L}{\partial y}=g, the gradient sent to uu is: