MTH 154, Calculus I

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4.1-Maximum and Minimum Values

State the Extreme Value Theorem (EVT) and produce esamples (algebraic and graphical) of where the conclusion of the EVT fails.

State Fermat's Theorem.

Locate critical values of functions given algebraically or graphically.

Locate the maximum and minimum values of a continuous function over a closed and bounded interval.

Explain the difference between a local extremum and an absolute extremum of a function.

4.2-The Mean Value Theorem

Know and state Rolle's Theorem and the Mean Value Theorem (MVT).

Use Rolle's Theorem and/or the MVT to obtain upper limits on the number of solutions certain equations may have. This is connected to the process of using the Intermediate Value Theorem to obtain rough locations and lower limits on the number of solutions to certain equations. Together both provide means to determine the precise number of solutions certain equations have. (See Section 4.9 on Newton's Method for further developments.)

4.3-How Derivatives Affect the Shape of a Graph.

Determine the maximal intervals over which a function is increasing/decreasing or concave up/concave down.

Apply the first and/or second derivative tests to determine whether a critical point of a function is a local maximum or a local minimum point of the function.

Produce an example where the second derivative test fails while the first derivative test secceeds.

Given information abouth the graph of a function, such as where the first derivatives and second derivatives of a function are positive/negative, various limits of the function and of its first derivative, etc., sketch the graph of the function.

Given the graph of the derivative of t function, determine where the function is increasing/decreasing or concave up/concave down.

4.4-Indeterminate Forms and L'Hospital's Rule

Apply L'Hospital's Rule to find limits that fit the indeterminate forms 0/0 and ∞/∞.

Express limits that fit the indeterminate forms 0⁰, ∞⁰, 1^∞, ∞-∞, etc. to the 0/0 and ∞/∞ forms and apply L'Hospital's Rule (or whetever applies) to find these limits.

Prove (by producing two appropriate examples) that these indeterminate forms really are indeterminate.

4.7-Optimization Problems.

Given an applied optimization problem, build an appropriate function with implied domain, identify whether the function is to be maximized or minimized, and then optiizing the function. This entails finding the critical points of the function in its domain and applying the appropriate sufficiency test (global first or second derivative tests, or the interval mehtod) to justify that the optimum solution has been found.

Explain why the local and global first and second derivative tests work in terms of graphical items such as increasingness and decreasingness of appropriate functions.

4.9-Newton's Method

Apply Newton's Method to solve equations.

Explain how Newton's method works graphically, and give a graphical example of how Newton's Method can cycle.

Use the IVT and Rolle's Theorem to determine the precise number of solutions of certain equations and obtain rough estimates of the solutions and to find all solutions using Newton's Method

4.10-Antiderivatives.

Find antiderivatives of functions using the formulas in Table 2 on p. 357.

Solve differential equations where a derivative (first or higher order) is given explicitly and sufficient initial data is given.

Explain why 2 initial conditions are needed to completely determine the solution when the second derivative is given explicitly, and so on.

Given acceleration data and location date, determine information such as maximum range, stopping distances, etc.