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 \(L_{(a,b)}(x)\).