Final answer:
To optimize a function s(x) using calculus 1 techniques, we must take its derivative to find maximum and minimum values, set it to zero to find critical points, and integrate to find the accumulated value. Sketching the function can aid in visualizing important features like critical points.
Step-by-step explanation:
To find a function s(x) that needs to be optimized, we will use calculus 1 techniques such as taking derivatives to find maxima and minima, setting derivatives equal to zero, solving for x, and integrating.
a) Maximize s(x)
To maximize s(x), we first take the derivative of s(x), set it equal to zero to find critical points, and then test these points to determine if they are indeed maxima by using the second derivative test or analyzing the behavior of s'(x) around the critical points.
b) Minimize s(x)
The process of minimizing s(x) is similar to that of maximizing it. After finding the critical points where the first derivative is zero, use the second derivative test to confirm that the points correspond to minima.
c) Solve for x in s(x) = 0
To solve for x in s(x) = 0, we simply set the function s(x) equal to zero and solve for the x-values which make the equation true.
d) Integrate s(x)
Integrating s(x) involves finding the antiderivative of s(x), which gives the accumulation of s(x) along its domain. Don't forget to add the constant of integration, which can be determined if initial conditions are given.
To apply these calculus 1 techniques successfully, it is often helpful to sketch the function and/or its derivative to visualize critical points, intercepts, and intervals of increasing and decreasing function values.