Calculus A Chapter 4B Flashcards

Flashcards for Calculus A Chapter 4B review.

Newton's Method Formula

Given a function $f(x)$, the iterative formula to find the root $x<em>{n+1}$ is: $x</em>{n+1} = x<em>n - \frac{f(x</em>n)}{f'(x_n)}$. Use this to approximate the zeros of a function by finding the zeros of its linearizations.

Calculator Use for Optimization

Graph $f(x)$ and $f'(x)$. Find roots/zeros/x-intercepts of $f'(x)$. These correspond to potential max/mins of $f(x)$. Verify using the second derivative test or by checking intervals around critical points.

Related Rates Problem Solving

1. Identify variables and rates of change. 2. Find an equation relating the variables. 3. Differentiate with respect to time $t$. 4. Substitute known values and solve for the desired rate. For example, if $V = \frac{4}{3}\pi r^3$, then $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

Local Linearization Formula

The tangent line approximation at a point $a$ is given by: $L(x) = f(a) + f'(a)(x-a)$. Use this to approximate the value of $f(x)$ for $x$ near $a$.

Calculating "dy"

Given $y = f(x)$, find $dy$ using $dy = f'(x)dx$. Evaluate $dy$ for specific values of $x$ and $dx$. For example, if $y = x^2$, then $dy = 2xdx$.

Optimization Steps

1. Define the function to be optimized and any constraints. 2. Find critical points by setting the derivative equal to zero. 3. Use the first or second derivative test to classify critical points as maxima or minima. 4.