Calculus 1: Section 4 Study Sheet

For a function to have both an absolute maximum and minimum point, what must be true?

The function must be continuous over a closed interval [a,b], m\le f(x)\le M . This is called the Extreme Value Theorem.

If the function is continuous over an open interval, what must be true?

The function may or may not have and absolute extrema.

Local Extrema

Minimum/maximum points that are not absolute extrema; These values are not the smallest/largest points on a function.

Critical Points

The points on a given function f where the value of the derivative f’ is either zero or undefined. If the point is outside the given interval or the derivative is a value only, the function has no critical points.

How are the absolute minimum and maximum determined?

First, find the critical points by solving for the derivative, setting the result to zero and solving for x. Then, plug in the values of a and b into the original function. Finally, determine the highest/lowest points. Those will be your absolute maximum/minimum.

Mean Value Theroem

If y=f(x) is differentiable over an open interval (a,b) in the function’s interior and is continuous over the closed interval [a,b], then c exists on the interval (a,b):

f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}

How can c be found to verify the Mean Value Theroem?

Start by finding the derivative of the given function. Then, replace x with c and determine each value of \frac{f(b)-f(a)}{b-a} =f^{\prime}(c) . Finally, solve for c.

In the equation f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} , what does c represent?

The point at which the secant line and tangent line have an equal slope (i.e. they are parallel on a graph).

Rolle’s Theorem

If a function y=f(x) is continuous on the interval [a,b] and differentiable over all values in the function’s interior (a,b), there exists a point c in (a,b) where f(C)=0 if f(a)=f(b).

What are some fundamental principles of the antiderivative?

An antiderivative f(x) must be equal to some constant, represented by c, if f’(x)=0, based on the Constant Multiple Rule.

An antiderivative f(x) of multiple terms, such as ax²+bx, can represent all the functions that satisfy the derivative f’(x) by adding c to the function’s, which represents any constant.

What must be done to find the function of the derivative f’(x) that passes through an exact point (x,y)?

Find the antiderivative, input x and y for their respective values, them solve for c to find the constant.