Study Guide 2

Study Guide: Test #2

Calculus I, Sr. Barbara Reynolds

Study Guide available: Week 7, after class on Day 20

Test Date: Week 8, Day 22

This test will cover material in Chapters 1 – 4.2. 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. As before, you are to do the first page of the test without a calculator. Once you turn in the first page, you may use a calculator during the rest of this test.

Identifying the shapes of particular function graphs:

Without using a calculator, you should be able to match the following functions to their graphs, where `a, b, c,` and `r` are constants:

`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=asqrt(x+b)`
`y=ax^3+b`		`y=a\|x\|+b`	`y=asqrt(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 text[exp](x)+b`

Functions:

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 [for example, evaluating `f(Delta x)` for a particular function]
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`]
Set up an expression for a difference quotient, `[f(x + h) – f(x)]//h,` and simplify this to the point where `h` does not appear alone as a factor in the denominator.
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
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 two functions graphically, you should be able to estimate the sum, difference, product, or quotient of the functions, and/or sketch a graph of this new function.

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

Why is the graph of the inverse of a function the reflection of the graph of the original function across the line `y = x?`
How are the slope of a linear function `f` and the slope of the inverse of `f` related? Remember that the inverse of a function is not necessarily perpendicular to the original function. The process of taking the inverse is to interchange the roles of the independent variable (that is, the “`x`”) and the dependent variable (that is, the “`y`”). What effect will this have on the slope?

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.`

Construct a table representing the inverse of a function for a function given in a table
Given a function represented in a table, you should be able to calculate estimated values for the difference quotient from the values in the table, and use these values to estimate rates of change.

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 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).`
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 Problem #8 from Test 1.)
You should be able to use a position function, `s(t),` to estimate or calculate the velocity, `v(t),` and the acceleration, `a(t).` You should be able to reverse this process; that is, starting with an acceleration function, `a(t) = g` and some initial conditions, you should be able to calculate the velocity function, `v(t),` and position function, `s(t).`

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 appear to be similar to 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 and 2.4. 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)`
Special exponential function, `y = f(t) = e^t= text[exp](t)` or `y = f(t) = e^(kt) = text[exp](kt)`
General special exponential function, `y = f(t) = A e^(kt) = A text[exp](kt)`

Solve continuous growth problems. That is, you should be able to set up an appropriate differential equation to model the problem situation, determine the initial conditions, and solve the problem. (See the exercises and problems at the ends of Sections 2.4 and 2.5.)
Given a differential equation of the form `dy//dt = ky` or `dy//dt = kt,` you should be able to determine whether the function `y(t)` will be an exponential function or a power function.

You should be able to solve a differential equation of the form `dy//dt = ky` or `dy//dt = kt` by finding three or more functions which satisfy the given differential equation.
You should be able to solve a differential equation with initial value by using this initial value to find the unique function that satisfies the given the differential equation and goes through the specified point.
You should be able to sketch a graph of the particular solution that you find using this process. That is, you should be able to sketch a graph of a power function or an exponential function that satisfies the given differential equation and goes through the specified point.

Differential Equations and Initial Value Problems

We have investigated two types of differential equations: `dy//dt = ay` and `dy//dt = at,` where `a` is a constant. One of these leads to an exponential equation, and the other to a power equation.

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 which satisfies the initial value condition.

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 that 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
Given a set of data in a table form, along with scatter plots of this data in three forms (`xytext[-plot],` `xln(y)text[-plot],` and `ln(x)ln(y)text[-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 (using `ln` or `text[exp]`) to get the function that models the original data. This is exactly the same process you had to use in developing your model for the AIDS data.
You should be able to use methods of calculus to solve problems:
- Cooling Body Problems: See Section 3.2.
- Falling Body Problems: See Section 3.3.

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).

The derivative can be used to optimize a function – by helping us to find the local maxima and minima. 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.

Given a graph showing both a function and its derivative, you should be able to determine which is which by considering the local maxima and local minima, and where each function crosses the x-axis.

Suggested Study questions for this test:

Certainly go over any problems you found difficult on the first test. You can expect to see a couple of those problems return on this test. (Mathematics is a cumulative discipline; thus, all mathematics tests are cumulative.)

Throughout Chapter 1, the authors kept coming back to several problem scenarios: asking whether the situation could be modeled by or represented by a function, what the independent and dependent variables would be, and what kind of expression might model the problem situation, and so on. Some of these problem scenarios reappear in Chapter 3. Pay attention to these questions, and try to predict where the authors are going with these threads.

Many of the Exercises in Chapters 2, 3, and the first two sections of Chapter 4 make good test questions. Look especially at those Exercises that build on the problem scenarios of Chapter 1. In particular, study the following problems:

Section 2.3: taking derivatives of polynomials

Section 2.5: any of the exercises for which I wrote out the solutions

Section 2.6 Exercises #5 – 6

Section 3.1 Exercises #1 – 9

Section 3.2 Exercises #1 – 3

Section 3.3 Exercises #3 – 5

Section 4.1 Exercises #4 – 6

Section 4.2 Exercises #1 – 6

You can use Maple to check your computations of derivatives. The Maple syntax is as follows:

f:=x->3*x-x^3;
f1:=diff(f(x),x);
f2:=diff(diff(f(x),x),x);