# MATH 260—Week 3 Lab Latest 2015

Math 260 – Week 3 Lab Name:

In calculus, much effort is devoted to determining the behavior of the graph of a function over an interval on the Cartesian Plane.

Finding x & y intercepts,asymptotes, intervals ofincreasing or decreasing,local maximum and local minimum points, concavity/curvature can all be done using Algebra and Calculus.

Using Algebra: ·The x and y-intercepts can be found by letting y = 0 then x = 0 respectively.

·Finding the zeros of the denominator of a function givesvertical asymptotes.

Using Calculus: ·The limit of the function as xà infinity will givehorizontal asymptotes.

·The zeros of the first derivative helps to find critical numbers in order to determine

intervals of increasing or decreasing and to find the max and min of afunction.

·The zeros of the second derivative helps to find inflection numbers in order to

determine Intervals of upward or downward concavity.

Graphing with Calculus

1.) The graph of is above. If you did not have this graph to look at but

wanted to sketch it, name at leastthree points on the graph and at leasttwo other features of

the graph that would be valuable to know in order to sketch an accurate graph.

Answer the questions 2-7 about some of the points and characteristics of a graph like the one above.

2.) How do you find x and y intercepts ?

3.) How can the first derivative help you to graph a function ?

4.) What is the value of the slope at a local maximum or minimum point on a curve ?

5.) How can the second derivative help you graph a function ?

6.) How do you find vertical and horizontal asymptotes ?

7.) Find the first derivative and the critical numbers for ? Then fill in the

blanks with the correct information regarding intervals of increasing or decreasing.

Critical Numbers:

Test Intervals:

Test Numbers:

Sign of f’(test #):

Increasing or Decreasing ?

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8.) Using the results from the problem above, determine the ordered pair point for the maximum

and minimumof f(x).

9.) Find the second derivative of then fill in the blanks with the correct

information regarding concavity.

Inflection Numbers:

Test Intervals

Test Number

Sign of f’’(test #)

Concave Upward or Downward ?

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Optimization:A method for finding the maximum or minimum of a quantity subject to a set of restraints or initial conditions.

To Optimize a Formula or equation, find the critical numbers for the formula. But the formula must be a single variable equation.

This often includes combining two or more equations into a single equation in a single unknown.

For example, combining the Area formula and the Perimeter formula for rectangles:

For example, if you solve the Area formula, A = LW, for Length,L =A/W, then the

Perimeter formula,P = 2L + 2Wbecomes P = 2(A/W) + 2W, and equation in one unknown.

If given a set of restraints or initial conditions, the zeros of the derivative of the resulting single variable formula will give a maximum or minimum with respect to that variable

10.) A rectangular garden will be fenced off with 220 feet of available material. What is the largest

area that can be fenced off ?

a.) What two formulas for rectangles are related in this problem ?

b.) Which formula do we want to optimize ? (This will be the primary equation)

c.) Write the primary equation so that it contains only one variable. Hint: Make use of the other

equation and fill in any given values.

d.) Find the derivative of the primary equation written in step c.) above then set it equal to zero and

solve for the unknown variable. What information do you now have?

e.) What is the largest area that can be fenced off ?

11.) The altitude of a U.S. Navy Flight Demonstration Squadron, the Blue Angels, jet that goes into

a dive and then again turns upward is given by h(t) = 2t3 – 25.9t2 + 1110 , h in ft., t in sec. Set

your graphing window to x-min = -1, x-max = 15, y-min = -10, and y-max = 1500 then use the

graph to answer the following questions for interesting facts about the Blue Angels go to

.blueangels.navy.mil/”>http://www.blueangels.navy.mil/

a.) What is the altitude of the jet at minimum point of the dive ?

b.) What is the maximum altitude between 2 and 5 seconds ?

c.) What is the altitude at 1 sec. and at 12 sec. ?

d.) What is the altitude at the point of inflection before the dive ?

Tangent and Normal lines

A tangent line touches a curve at a single point. At that point, the tangent line and the curve are sloping in the same direction.

A Normal line is perpendicular to the tangent line.The normal line is used in determining direction for certain forces and when working with physical laws, like those for gravity or reflection. They are also used in computer graphics to determine orientation towards a light source in order to mimic a curved surface on a flat medium.

To find the equation of the tangent line, find the slope by finding the first derivative and substitute-in the x-coordinate of a given point to find the slope at that point. Then use the point, the slope, and the formula y – y1 = m(x – x1) to find the equation.

To find the equation of the normal line, use the same process except use a slope that is the opposite reciprocal of the slope of the tangent line.

12.) Find the equation of the Tangent line and the equation of the Normal line at (2, -1) for

f(x) = 2×3 – 5×2 + x + 1

An application of the tangent and normal lines in AC circuits:

Bandwidth of a Series Resonance Circuit

Series Resonance circuits are one of the most important circuits used in electrical and electronic circuits. They can be found in various forms of applications such as AC mains filters, noise filters and also in radio and in the past, television tuning circuits producing a very selective tuning circuit for the receiving of the different frequency channels.

Given a curve that maps Current versus frequency, thetangent line that has a slope of zero represents the currentmaximum IMAX and the horizontal line that goes through theinflection points reveal the values of the lower and upper frequencies at – 3dB or 0.707 IMAX

Further, the line of symmetry that goes throughfr isnormalto the tangent line atIMAXand the two parallel normal lines atfL and f H help to determine the frequency difference which is called the Frequency bandwidth.

http://www.electronics-tutorials.ws/terms

13.) If the following equation represents the curve for a frequency bandwidth similar to the one

above, find the maximum current, the inflection points, then give the equation of the vertical

lines that go through each inflection point. Determine the band width. Round all values to the 100ths

xmin= -2, xmax = 3, ymin= -2, ymax = 3

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