Unit 3 Summary

The Derivative

I can…

  • Define what a derivative is.
  • Interpret the derivative \(f'(x)\) as
    • a rate of change
    • a measure of the sensitivity of output to changes in input
    • the slope of a tangent line.
  • Estimate the derivative using a table of data.
  • Estimate the derivatives from the plot of a function and its tangent line.
  • Estimate the derivative \(f'(x)\) at \(x=a\) using the average rate of change. \[ f'(a) = \frac{f(a+h)-f(a)}{h} \] where \(h\) is a very small number
  • Use desmos to find an approximation of the derivative of a function to a desired accuracy.
  • Use the derivative to determine whether a function is increasing or decreasing.
  • Define marginal cost.
  • Calculate the marginal cost of \(C(x)\) at \(x=a\) using the formula \(C(a+1)-C(a)\)
  • Explain why marginal cost is an example of an approximate derivative
  • Explain how the derivative connects displacement, velocity and acceleration.

The Second Derivative

I can…

  • Give the definition of the \(k\)th derivative \(\displaystyle{\frac{d^kf}{dx^k} = f^{(k)}(x)}.\)
  • Explain why acceleration \(a(t)\) is the second derivative of displacement \(s(t)\). That is, \(a(t) = s''(t).\)
  • Use the second derivative to determine the concavity of a function.

Partial Derivatives

  • Define what a partial derivative is.
  • Explain the difference between \(\displaystyle{f_x = \frac{df}{dx}}\) and \(\displaystyle{f_y = \frac{df}{dy}}\)
  • Estimate the partial derivatives from a table of data
  • Estimate the partial derivatives from a contour plot.
  • Use desmos to approximate the partial derivatives using \[ \frac{\partial f}{\partial x} \approx \frac{f(x+h,y) - f(x,y)}{h} \quad \mbox{and} \quad \frac{\partial f}{\partial y} \approx \frac{f(x,y+h) - f(x,y)}{h} \] where \(h\) is a very, very small number.

Local Linear Approximation

I can…

  • Create the linearization \(L_a(x)\) for function \(f(x)\) at \(x=a\) using the formula \[ L_a(x) = f(a) + f'(a) (x-a). \]
  • Approximate \(f(x)\) using the linearization \(L_a(x)\).
  • Create the linearization \(L_{(a,b)}(x,y)\) for function \(f(x,y)\) at point \((a,b)\) using the formula \[ L_{(a,b)}(x,y) = f(a,b) + f_x(a,b) (x-a) + f_y(a,b) (y-b). \]
  • Approximate \(f(x,y)\) near the point \((x,y)\) using the linearization \(L_{(a,b)}(x)\).