Study Guide - Final Exam

Study Guide: Final Exam

Calculus I, Sr. Barbara Reynolds

Exam date and time: insert DAY, DATE, TIME

The final exam is cumulative, covering material in Chapters 1 – 5 of our text. This will be an individual, in-class exam. Part A of this exam is to be done without a calculator. After turning in your work on Part A, you may use your calculator (even your TI-89) and one 3”´5” note card as you work on the rest of the exam.

Functions

a function represented as an algebraic expression, you should be able to set up a difference quotient, and simplify the difference quotient to the point where you do not have a factor of h by itself in the denominator
You should be able to set up and simplify a difference quotient for functions of the following types: `y = ax + b,` `y = ax^2 + b,` `y = ax^3 + b,` `y = a // (x + b),` `y = (a // x^2) + b,` and `y=sqrt(ax+b).`
Given a function represented as an algebraic expression, in a table of values, or graphically, you should be able to give its inverse using the same kind of representation. In other words, if the function is given in a table, you should be able to give the inverse in a table; if the function is given in a graph, you should be able to sketch a graph of its inverse; if the function is given as an algebraic expression, you should be able to give its inverse as an algebraic expression.
You should know the basic shapes of the graphs of the following functions, where a, b, and c are positive constants. You should be able to sketch graphs of these function (or match a function to its graph) without using a calculator. You should be able to sketch graphs of these as generic functions (that is, without having particular numbers of a, b, and c).

`y=ax + b`	`y=a f(x) + b`	`y=a sin(x) + b`	`y= (x+a) (x+b) (x+c)`
`y=ax^2 + b`	`y=a f(x + b)`	`y=a cos(x) + b`
`y=ax^3 + b`	`y=ax + b`	`y=a \|x\| + b`
`y=a/x`	`y=a/(x + b)`	`y=a \|x + b\|`	`y=a ln(x) +b`
`y=a^x`	`y=(a/x) + b`	`x^2 + y^2 = r^2`	`y=a e^x + b = a exp(x) + b`

You should be able to calculate first and second derivatives for any function in the above table.
You should be able to analyze the expression so that you can find interesting graphical features of the function such as `x`- and `y`-intercepts, regions where the function is positive or negative, regions where the function is increasing or decreasing, regions where the function is concave up or concave down.
You should be able use this analysis to sketch a graph of the function.
Given a graph containing several functions, you should be able to determine which graph is the function and which is its derivative.
Given the graph of a function, you should be able to sketch a graph of its derivative.

Difference Quotients and Rates of Change

You should be able to set up and simplify a difference quotient, and use the difference quotient to estimate the derivative of the function.
Given a function as a table of values, you should be able to apply the idea of setting up a difference quotient to find an estimate for the derivative of the function. That is, given a table of values for `x` and `y,` you should be able to calculate `text[(]y_2 – y_1text[)] //text[(]x_2 – x_1text[)],` and explain why this calculation gives an approximation to the derivative.

Mathematical Modeling and Problem Solving

You should be able to use methods of calculus to solve problems:
Developing a mathematical function to represent data (AIDS data project, see Chapter 2)
Recall that in the AIDS project, the strategy was to decide which family of functions to use by looking at log-log and semi-log graphs of the data.
Cooling Body Problems: See Section 3.2. Recall that in Section 3.2, the strategy involved using a change of variable (substituting `y = T – 21` in the equation) to change the equation from one we didn’t know how to solve to one that we know how to solve. This method was used again in Section 5.2 (Stokes' Law).
Falling Body Problems: See Sections 3.3 and 5.1.
Optimization Problems (that is, finding maxima and minima): See Chapter 4.
Finding the rate of change in the length of the hypotenuse of a right triangle when one or both of the legs of the triangle is changing with time: See the Air-Traffic Control problem and the problem about the baseball diamond on Test 3.

Given a problem situation expressed as a word problem or in a table of data, you should be able to demonstrate the following skills:

Identify independent and dependent variables in the situation, and give an appropriate domain for the independent variable
Sketch a graph showing the relationship between two of the variables in the situation
Read a graph which represents a problem scenario, find particular values in the graph, and interpret the graph in the context of the situation
Develop an algebraic expression for a function which models a linear situation, and explain what the slope (rate of change) means mean in the context of the problem
Given a set of data in a table form, along with scatter plots of this data in three forms [`xy`-plot, `x`-`ln(y)`-plot, and `ln(x)`-`ln(y)`-plot], you should be able to choose the graph which appears most nearly linear, construct an appropriate linear equation, and use this to determine an appropriate function to model the data. (See Section 2.6 and your project on the AIDS data.)

To do this, you will need to use the appropriate graph along with the given data to estimate the slope and `y`-intercept of the line.
Once you have an expression for the line, you may need to transform it to get the function that models the data.Once you have an expression for the line, you may need to transform it to get the function that models the data.

Differential Equations and Initial Value Problems

We have studied three types of differential equations: `dy//dt = ay` and `dy//dt = at,` where `a` is a constant, and `d^2y//dt^2 = -k^2 y,` where `k` is a constant. The first of these leads to an exponential function (`y = A e^(kx)`), the second to a power function `[y = (k//2) x^2 + C],` and the third to a circular function [`y = A sin (k x) + B cos(k x)].` The initial conditions give the information needed to calculate the parameters for the function. Differential equations are used to solve problems like the Cooling Body problem (Section 3.2), the Falling Body problem (Section 3.3), and the Periodic Motion (Sections 5.3 and 5.4).

You should be able to solve a differential equation by finding a family of functions for which the given differential equation is true.
You should be able to solve a differential-equation-with-initial-value problem by finding the particular function for which the given differential equation is true, and satisfies the initial value conditions.

Calculating Derivatives

You should be able to calculate derivatives by hand using the rules:

Power rule, the sum and difference rules, and the general power rule
Product rule
Chain rule
Implicit differentiation
Derivatives of each of the trigonometric functions
Quotient rule (Problem 10, and the end of Section 4.7)

There will be a problem asking you to find the derivative in Part A of the test (which you are to do without a calculator). There are a lot of good practice problems for calculating derivatives in the WeBWorK problem sets.

Using Derivatives to Solve Problems

The derivative always tells us about a rate of change. Depending on the context of the problem the rate of change may be

the slope (of the graph of the function),
the velocity (rate of change in position), or
the acceleration (rate of change in velocity).

Essentially, setting the derivative equal to `0` helps us to find places where the graph has a horizontal tangent. The derivative can be used to optimize a function – since there will be horizontal tangents at the local maxima and minima points. That is, a function can reach a local maximum or minimum at a point where the slope of the graph is `0` (so that the tangent line is horizontal). A function reaches its global maximum or minima at a local maximum or minimum, or at one of the endpoints of the interval under consideration.

You should be able to calculate the first and second derivatives of a given function.
You should be able to find the zeros (or the roots) of a function, its derivative, and its second derivative.
You should be able to use the function and its first and second derivatives to find critical values of the function, and use these to find maxima, minima, and points of inflection.
You should be able to determine when the function is increasing or decreasing, and when the graph is concave up or concave down.

Which application problems or projects that we have studied this semester have helped you develop critical thinking and analytic reasoning skills that will help you to be a better citizen?

Suggested Study Problems for the Exam

Review problems from the three tests we’ve had this semester, paying special attention to those problems with which you had difficulty. All mathematics tests are cumulative. If there was a problem on an earlier test that several students found difficult, it may reappear in a slightly different form on the exam.
I will use the problems presented in the Examples, Checkpoints, and Activities of Chapters 1 – 5 of our text as well as problems in the Web Work problem sets as inspiration as I develop questions for this Exam.
Here are some particularly interesting problems to use for your review:

The solutions to interest rate and population growth problems in Sections 2.4 and 2.5; these are differential-equations-with-initial-value problems, and there are some good examples in the Activities, Examples, and Checkpoints of these sections
Position-velocity-acceleration problems where gravity is a constant; see Sections 3.3 and 5.1
The Air Traffic Control project and the problem about the baseball diamond on Test 3; in these problems, one or both legs of a right triangle change with respect to time; that it, its length is a function of time. In order to take the derivative, you needed to use the chain rule to solve these problems.
The method of substitution of variable that was used in the solution to the Cooling Body problem (Section 3.2.3), and the solutions to the Activities of Section 5.1.2 (in the discussion of Stokes' Law)