4.1 Related Rates
- To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.
- In terms of the quantities, state the information given and the rate to be found.
- Find an equation relating the quantities.
- Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.
- Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.
4.2 Linear Approximations and Differentials
- A differentiable function can be approximated at by the linear function
- For a function if changes from to then
is an approximation for the change in The actual change in is
- A measurement error can lead to an error in a calculated quantity The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by
- To estimate the relative error of a particular quantity we estimate
4.3 Maxima and Minima
- A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
- If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
- A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.
4.4 The Mean Value Theorem
- If is continuous over and differentiable over and then there exists a point such that This is Rolle’s theorem.
- If is continuous over and differentiable over then there exists a point such that
This is the Mean Value Theorem.
- If over an interval then is constant over
- If two differentiable functions and satisfy over then for some constant
- If over an interval then is increasing over If over then is decreasing over
4.5 Derivatives and the Shape of a Graph
- If is a critical point of and for and for then has a local maximum at
- If is a critical point of and for and for then has a local minimum at
- If over an interval then is concave up over
- If over an interval then is concave down over
- If and then has a local minimum at
- If and then has a local maximum at
- If and then evaluate at a test point to the left of and a test point to the right of to determine whether has a local extremum at
4.6 Limits at Infinity and Asymptotes
- The limit of is as (or as if the values become arbitrarily close to as becomes sufficiently large.
- The limit of is as if becomes arbitrarily large as becomes sufficiently large. The limit of is as if and becomes arbitrarily large as becomes sufficiently large. We can define the limit of as approaches similarly.
- For a polynomial function where the end behavior is determined by the leading term If approaches or at each end.
- For a rational function the end behavior is determined by the relationship between the degree of and the degree of If the degree of is less than the degree of the line is a horizontal asymptote for If the degree of is equal to the degree of then the line is a horizontal asymptote, where and are the leading coefficients of and respectively. If the degree of is greater than the degree of then approaches or at each end.
4.7 Applied Optimization Problems
- To solve an optimization problem, begin by drawing a picture and introducing variables.
- Find an equation relating the variables.
- Find a function of one variable to describe the quantity that is to be minimized or maximized.
- Look for critical points to locate local extrema.
4.8 L’Hôpital’s Rule
- L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form or arises.
- L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form or
- The exponential function grows faster than any power function
- The logarithmic function grows more slowly than any power function
4.9 Newton’s Method
- Newton’s method approximates roots of by starting with an initial approximation then uses tangent lines to the graph of to create a sequence of approximations
- Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers does not approach a finite value or it approaches a value other than the root sought.
- Any process in which a list of numbers is generated by defining an initial number and defining the subsequent numbers by the equation for some function is an iterative process. Newton’s method is an example of an iterative process, where the function for a given function
- If is an antiderivative of then every antiderivative of is of the form for some constant
- Solving the initial-value problem
requires us first to find the set of antiderivatives of and then to look for the particular antiderivative that also satisfies the initial condition.