Study Guide 1

Study Guide: Test #1

Calculus I, Sr. Barbara Reynolds

Study Guide available: Week 4, Day 9 of the semester

Test Date: Week 4, Day 11 of the semester

This test will cover material in Chapter 1 through Section 1.5, and Chapter 2 through Section 2.4. Although you will be expected to do some computations during the test, the problems will focus on conceptual understanding of functions, difference quotients, and the process of modeling problem situations with mathematical functions.

This will be a closed book, individual test.

The first page of this test will involve some problems that you should be able to do without a calculator; once you turn in the first part of the test, you may use a calculator. Of course, you should still be able to do all the computations that are listed on the Study Guide for the Quiz.

The remainder of the test will have 5 – 6 problems. You may use a calculator during the second part of the test. You will be asked to show your work and/or to explain your reasoning.

Sketch Graphs and Linear Equations:

You should be able to match the following functions to their graphs (without using a graphing calculator), where `a, b, c` and `r` are assumed to be positive constants:

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

For each of these functions presented as sketch graphs, you should be able to determine the effect of changing the sign of the constants from positive to negative.
Given some information about a line – two points, one point and the slope, the `x-` and `y-text[intercepts],` a point and the equation of a line that is either parallel or perpendicular to it – you should be able to write an equation for the line.

Functions:

A function is a pairing of the values of one varying quantity with the values of another varying quantity in such as way that each value of the first variable is paired with exactly one value of the second variable.

A function may be expressed verbally in the context of a problem situation, numerically (in a table of values), visually (in a graph), or algebraically (as an algebraic expression).
A function may be thought of as a process that converts inputs to outputs; it may also be thought of as an object with specific properties.

Given a function, f, expressed algebraically (by an explicit formula), visually (in a graph), or numerically (in a table), you should be able to

Evaluate it at specified values, estimating the value if necessary
Identify the independent and dependent variables in the problem situation
Estimate values of the independent variable that will give specified results (for example, find the value of `x` such that `f(x) = 3)`
Determine whether a function is additive or multiplicative, and use the algebraic calculation given in the definitions of these properties to support your reasoning
Determine whether a function is even or odd, and use the algebraic calculation given in the definition of these properties to support your reasoning
Explain what the property of being even or odd means graphically
Use the property of even or odd, to complete the graph of a function. That is, given a sketch graph of a portion of a function and whether it is even or odd, you should be able to complete the sketch graph of the function.
Build a new function from given functions by adding, multiplying, subtracting, or dividing

Given algebraic expressions for `f` and `g`, you should be able to find an algebraic expression for `f+g,` `f–g,` `f times g,` and `f//g`.
Given algebraic expressions for functions f and g, you should be able to find an algebraic expression for their composition, `(f@g)(x)=f(g(x)).`
Given two functions graphically, you should be able to estimate the sum, difference, product, or quotient of the functions, and sketch a graph of this new function.

Sketch a graph of the inverse of a function, and evaluate the inverse at specified values

The graph of a function and its inverse are reflections in the line `y = x.`
How are the slope of a linear function `f` and the slope of the inverse of `f` related?

Given an expression for a function `y=f(x),` you should be able to find an expression for each of the following:

Find an expression for the inverse of the function by interchanging the variables `x` and `y` and solving for `y.` Note that in this context, `f^(-1)(x)` is the inverse – not the reciprocal – of the function `f.`
Find an expression for the reciprocal, `f(x)^(-1).`

Given a table of values representing the relationship between two variables. You should be able to do the following:

Sketch a scatter plot of the relation.
Determine whether the relation given in the table is a function.
Construct a table representing the inverse of the relation, and determine whether this inverse is a function.

Difference Quotients and Rates of Change:

Difference quotients are used to estimate and calculate rates of change.

For a function given as an algebraic expression, you should be able to set up a difference quotient to represent the rate of change.

That is, you should be able to set up (and simplify) an expression for `(f(x+h)-f(x))/((x+h)-x)=(f(x+h)-f(x))/x.` You may use `h` or `Deltax` to denote the “small change” in the independent variable.
You should be able to simplify this expression to the point where the “small change” (whether denoted by `h` or `Deltax`) does not appear by itself in the denominator.

You should be able to interpret this rate of change in the context of the given scenario using appropriate units. For example, if `f` gives the height (in feet) of an object at time `t` (in seconds), the difference quotient will give a change in height with respect to time (in feet per second).
For a function given as a table of values, you should be able to calculate values of the difference quotient to estimate the rate of change. (E.g., see Activity 2, in Section 2.1.)

When a function has the property of local linearity at a point, `c,` and we zoom in repeatedly on the graph of the function around the point `c,` the function will appear to be straight around this point. In this case, the graph of the function will approximate the graph of the tangent line at this point.

You should be able to use a difference quotient to find the slope of this tangent line.
You should be able to estimate the value of the derivative using approximate tangent lines to the graph at a particular point.
You should be able to use the special derivative rules that are developed in Section 2.3. In particular, you should be able to find derivatives of the following kinds of functions:

Constant function, `y=f(x)=k`
Linear function, `y=f(x)=kx`
Quadratic function, `y=f(x)=kx^2`
General power function, `y=f(x)=kx^n`
Constant multiple of a function, `y=kf(x)`
Sum of two functions, `y=f(x)+g(x)` or `y=kf(x)+cg(x)`
General polynomials (using the general power rule, the constant multiple rule, and the sum rule)
Exponential function, `y=f(t)=b^t` or `y=f(t)=b^(kt)`
General exponential function, `y=f(t)=Ab^(kt)`

Mathematical Modeling:

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 that models a linear situation, and explain what the slope (rate of change) means in the context of the problem

Examples of the kinds of problem situations that our authors have presented throughout the text: scatter plot of data (cigarette, newspapers, teacher’s salaries), postage rates (step function), learning curve, distance and speed (speed is a rate of change), speed of a falling body (modeled by a quadratic function), car payments (giving an example of a function and its inverse), population growth, compound interest (modeled by an exponential function)

Good Study Questions:

The Study Guide for the Quiz
Activities and Checkpoints throughout the text
Exercises at the end of each section of our text, and the corresponding WeBWorK problems
Class notes particularly for the computation of difference quotients for a given function, and for sketch graphs of functions