MTH 154, Calculus I

In the following objectives, some mathematical notation is uses. Depending on what character set is used or is available with your browser, some of these notations may or may not come out as intended.

---

3.1-Derivatives of Polynomials and Exponential Functions

Know and state the specific differentiation formulas (for constants, power functions, and exponential functions)

Know and state general differentialtion formulas (constant multiple rule, sum/difference rules).

Efficiently find derivatives of functions built as sums/differences of multiples of power functions and exponential functions.

Locate points of horizontal tangency of functions of the type in the previous objective.

Determine if a piecewise function is continuous and /or differentiable at a knot point.

Note: Once differentiation formulas and their usage are mastered, a general objective is that the student should be able to determine graphical information such as location of points of horizontal (or vertical) tangency of a curve, equations of tangent and normal lines to a curve, location on a curve for which the tangent line passes through a specified point, etc.

3.2-The Product and Quotient Rules

Know and state the general formulas (product and quotient rule).

Explain to a fellow student why the derivative of a product of two functions is not the product of the derivatives of the two functions and why the derivative of the quotient of two functions is not the quotient of the derivatives of the two functions.

Apply all differentialtion formulas thus far to efficiently find derivatives of functions built as sums/differences/products/quotients of multiples of power functions and exponential functions.

3.4-Derivatives of Trigonometric Functions

Derive lim_x→0sinx/x=1 and lim_x→0(1-cosx)/x=0, and find related limits using these.

Derive the derivatives of all six trigonometric functions.

Efficiently find derivatives of functions constructed in the form of sums/differences/products/quotients of multiples of power functions, exponential functions, and trigonometric functions.

3.5-The Chain Rule

Know and use both versions of the Chain Rule given on page 215.

Efficiently find derivatives of functions constructed in the form of sums/differences/products/ quotients/COMPOSITIONS of multiples of power functions, exponential functions, and trigonometric functions.

Explain to a fellow student why the derivative of (7x+1)⁶ is not equal to 6(7x+1)⁵.

Find derivatives of sums/differences/products/quotients/COMPOSITIONS of functions given by graph or table rather than just by formula.

Note: The chain rule is the most difficult of the differentiation formulas to master, and it is the source of most differentialtion errors. It is very important to master it because the chain rule will turn out to be your friend rather than an obstacle. It will allow you to solve some several important problems with relative ease, and it will be pervasive in all of calculus from here on.

3.6-Implicit Differentiation.

Explain to a fellow student the difference between a function given explicitly and a function given implicitly.

Use the chain rule to find derivatives of functions given implicitly.

Know and state the derivatives of the inverse sine, inverse tangent, and inverse secant functions.

3.7-Higher Derivatives

Find second and higher order derivatives of functions of the form sums/differences/products/ quotients/COMPOSITIONS of multiples of power functions, exponential functions, trigonometric functions, and the three inverse trigonometric functions cited above.

Given the graphs of a function of a function, its derivative, and its second derivative identify which is which.

Recognize patterns in finding higher order derivatives of functions.

Given equations of motion, determine when an object passes or strikes a specific location, reverses direction, etc. and determine velocities and accelerations.

3.8-Derivatives of Logarithmic Functions

Know, state, and derive the derivative of the natural logarithm function.

Explain to a fellow student why the power rule holds for all real exponents rather than just the integer exponents.

Apply logarithmic differentiation to efficiently differentiate complicated products of functions and functions raised to function powers.

3.10-Related Rates

For related rates problems, identify the rates known (as derivatives of quantities with respect to time or whatever is appropriate), identify the rates desired, establish relations relating the quattities involved, and apply differentiation with the chain rule to then determine the desired rates.

3.11-Linear Approximation and Differentials

Use differential notation and find differentials of the same functions one can differentiate.