Study Guide 3

Study Guide: Test #3

Calculus I, Sr. Barbara Reynolds

Study Guide available: Week 13, Day 36

Test Date: Week 13, Day 38

This test will cover material in Chapters 3 – 5. 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. (However, you may not have a note card for this test.)

Identifying the shapes of particular function graphs

Without using a calculator, you should be able to match function of the form `y = A sin(Bx + C) + D` or `y = A cos(Bx + C) + D` to their graphs, where `A, B, C,` and `D` are positive constants.

Study Questions:

See Activity 1, in Section 5.4.1.
What will happen to the graph of a function if three of these parameters are held constant, and the remaining one is doubled, or tripled, or halved?
What will happen to the graph of the function if either the `A` or the `D` is changed from positive to negative?

Given a function expressed graphically (and without knowing the algebraic expression for this function), you should be able to use the graph to

Estimate the zeros of the function [that is, the values of `x` where `f(x) = 0`]
Estimate the location of the zeros of the derivative [that is, the values of `x` where `f'(x) = 0]`
Estimate the location of the zeros of the second derivative [that is, values of `x` where `f''(x) = 0]`
Sketch a graph of the derivative function

Study Question:

See problem 5 on Test #2.

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.

Given a graph showing a function and its first two derivatives on the same axes, you should be able to identify which is the original function, which is its first derivative, and which is the second derivative. You should be able to explain how you make your choice.

Study Questions:

There are some of these in the WeBWorK problem sets. sets. For example, see Hwk_4f, problem 1.

Differential Equations and Initial Value Problems

We have investigated 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. One of these leads to an exponential equation, another to a power equation, and the third to function of the form `y(t) = A sin(t) + B cos(t).`

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.
Given a differential equation of one of the following forms, you should be able to identify from which family of functions the solution comes, and use the initial values to find the particular function which fits the particular initial value situation:

`dy/dt = ky,` with `y(a) = b`
`dy/dt = kt,` with `y(a) = b`
`(d^2y)/(dt^2) = -ky,` with `y(a) = b` and `y'(a) = c`

Study Questions:

The Exercises of Sections 3.2, 3.3, and 5.4

Applications which give rise to Differential Equations

Given a differential equation with information about initial value(s), you should be able to identify to which family of function the solution belongs. You might be given a differential equation from one of the following families:

`dy/dt = ky,` with `y(a) = b`
`dy/dt = kt,` with `y(a) = b`
`(d^2y)/(dt^2) = -ky,` with `y(a) = b` and `y'(a) = c`

The solutions to these differential equations will be power functions, exponential functions, or a linear combination of sines and cosines. You should be able to identify the family of functions, and then to solve for the parameters (that is, for the constants `k, n, A, B,` etc.).

The following applications problems have motivated our study of these differential equations. For each type of application, you should be able to

Sketch a diagram to represent the situation.
Recognize the family of differential equations that best represents the problem in the situation—whether the situation can best be modeled by an exponential function, a power function, or a cyclic function.
Find an expression for a differential equation that represents the situation.
Use information in the problem situation to determine initial conditions.
Simplify the equation and use the initial condition to solve the equation.
Explain what the solution means in the context of the given problem situation.

Population growth (Section 2.5, which also includes a discussion of what it means to solve a differential equation)
Cooling body (Section 3.2)
Falling body (Sections 3.3 and 5.1)
Systems that oscillate (Sections 5.3 and 5.4)

Calculation of derivatives

The Summaries at the ends of Chapters 3, 4, and 5 list the rules we have covered for calculating derivatives. In fact, the Summary at the end of Chapter 5 includes a cumulative list of the various rules for differentiation.

You should be able to

Use the short-cut formulas to calculate derivatives
Use the difference quotient to prove a derivative formula
Set up and simplify a difference quotient for sine or cosine function (I will provide the necessary angle sum formula on the test.)
Using the sum, product, and chain rules to find other derivative formulas
Express other trigonometric functions in terms of sine and cosine functions, and use these to develop formulas for the derivatives of the tangent, cotangent, secant, and cosecant functions.

Study Questions:

The Examples and Checkpoints in Sections 5.5 and 5.6.
Lots of the WeBWorK problems
Examples and Activities in Section 5.3.6
It will be helpful to know the unit circle and the two standard triangles.

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 the sine and cosine functions.
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).
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).`

Study Questions:

Problems on the previous tests

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.
The strategy for finding local maxima and local minima is to take a derivative, set it equal to zero, and solve for the independent variable.

Suggested Study questions for this test:

Certainly go over any problems you found difficult on the last two tests.
Go over your work on the two projects, and be sure that you understand how to solve those problems.
WeBWorK problems make good test questions.
Be sure that you understand how functions and their derivatives appear in graphs.
There are lots of exercises for calculating derivatives. You can use Maple to check your computations of derivatives. The Maple syntax is diff(f(x), x) and diff(diff(f(x), x), x).