Mr. Lovelace's Math Site

Applications of the Derivative

What are the Questions?

Find critical numbers.
Find intervals on which a function is increasing and decreasing using the sign of the first derivative.
Find relative and absolute extrema.
Use the first or second derivative test to decide whether the critical points (candidates) represent relative maxima or relative minima.
Find inflection points.
Find intervals on which a function is concave up or concave down using the sign of the second derivative.
Use limits to find all vertical and horizontal asymptotes.
Represent an optimization word problem in functional form.
Determine the quantity, P, to be maximized or minimized and identify the variables which are involved.
Draw a diagram, if possible, to illustrate the problem and list any other relationship(s) between the variables.
Identify local extrema, points of inflection, and intervals on which a function is increasing, decreasing, concave up, or concave down by investigating its first/second derivative.
Know the derivatives of polynomial, rational, power, trig., exponential, logarithmic, inverse trig. functions.
Use the established derivative rules (power, sum, product, quotient, chain) to find the derivatives of functions “built” from polynomial, rational, power, trig., exponential, logarithmic, inverse trig. functions.
Find an equation of a tangent line (or secant line) to the graph of a function.
Interpret the meaning of a derivative in a specific applied problem.
Use tangent lines to approximate more complicated functions.
Use derivatives to find extreme values of a given function.
Use implicit differentiation to find unknown rates of change by relating them to known rates of change.
Explain the difference between an absolute extreme point and a relative extreme point.
Given a function f continuous on a closed interval [a, b], find the values at which f takes on its absolute maximum and minimum values and find the values of extreme values.
Write the definition of critical points and explain their importance in finding relative and absolute extreme points.
Identify both the candidates for extreme values of a function and the extreme values of the functions, if relevant.
Explain what it means for a function to be increasing (decreasing) on an interval.
Given a function f, use the first derivative to identify the intervals on which f is increasing, decreasing.
Use the first derivative test to determine the nature (relative maximum, relative minimum, neither) of a critical point.
State the definitions of concave up and concave down.
• Use slopes of tangents to a graph to relate the concavity of a function to the increasing/decreasing nature of the first derivative.
Explain what it means for a point to be an inflection point and use analysis of the second derivative to identify inflection points.
Given a graph of a function f, sketch a graph of f' . Given a graph of f ' , sketch a graph of f.

Topic Notes/Cheat Sheets/Review Problems

Topic Videos

Extreme Values of Functions

Extreme Value Theorem ( EVT)

Connecting f and f" to f

The graphs of f(x), f’(x), f’’(x)

Related Rates

Optimization

Page updated

Report abuse