We acknowledge the University of Delaware's Morris Library for the use of its resources during the preparation of this text. Morrishas taught courses ranging from college algebra to calculus and statistics since 1987 at the Dover Campus of the University of Delaware.
SOLVING LINEAR EQUATIONS
Resolve fractions
Remove grouping symbols
Use addition (and/or subtraction) to have variable terms on one side of the equation
Divide the equation by the variable’s coefficient
As a check, verify the solution in the original equation
Solving linear equations in several variables simply sets aside a variable of interest. Move terms with y to one side of the equation and any remaining terms to the opposite side.
Read problems carefully
Identify the quantity of interest (and possibly useful formulas)
A diagram may be helpful
Assign symbols to variables and other unknown quantities
Use symbols as variables and unknowns to translate words into an equation(s)
Solve for the quantity of interest
Check your solution and whether you have answered the proper question
LINEAR EQUATIONS AND THEIR GRAPHS
In addition to the algebraic representation of linear equations used here, many applications use elegant matrix representations. The slope, m, of a line can be described in several ways; "increment divided by run", or "change iny, denoted by Δy, divided by change inx,Δx".
FACTORING AND THE QUADRATIC FORMULA
To determine anyx-intercepts (called "roots" or zeros of the quadratic), set=0 and solve 0=ax2+bx+c. Use the quadratic formula to determine the price below which producers will not produce a product whose supply function is S(p) =p2+10p−200.
FUNCTIONS AND THEIR GRAPHS
The graph below cannot be the graph of a function as it fails the vertical line test at x=a. The third and last segment of the graph is part of the horizontal line y=4 with an open circle at (1, 4).
SLOPES AND RELATIVE CHANGE
The denominator of a difference quotient is a factor of the numerator, so the difference quotient can be simplified. According to the Oxford Dictionary of the English Language (OED), the word Algebra became an accepted form among many similar forms in the seventeenth century.
SUPPLEMENTARY EXERCISES
SLOPES OF CURVES
The slope of the tangent to the curve at a point P, in this case a circle, is the slope of the curve at that point. The slopes of the tangents at different points of the parabola = x2 are twice the value of x there.
LIMITS
In this case, as x 2 approaches from the left (negative side), (towards the open endpoint) the limit is 4. In this case, as x 4 approaches from the left (negative side), along the horizontal line, (to almost 4), the limit is 3.
DERIVATIVES
As the distance between x and decreases, that is, asx→a, the secant approaches the tangent and becomes the derivative.
- DIFFERENTIABILITY AND CONTINUITY
- BASIC RULES OF DIFFERENTIATION Recall the power rule,
- CONTINUED DIFFERENTIATION
- INTRODUCTION TO FINITE DIFFERENCES
One type of cost per unit of time increases with increasing time (such as holding and storing inventory quantities), while another type decreases with increasing time (such as spreading overhead and fixed costs over a longer period of time). . Note: We will revisit this example in Chapter 3, where the derivative will be used to determine the optimal solution. If all three conditions hold, the function is continuous at x=a.. a) The function is a polynomial and is continuous everywhere.
The above example illustrates that for the derivative of the function [g(x)]r, the original exponent becomes a coefficient and the new exponent has been reduced by one and multiplied by the derivative of g(x), as previously shown in the shaded box. . Using the general power rule, there is no need to expand the algebraic expression to determine the derivative as in the previous example.
SUPPLEMENTARY EXERCISES
DESCRIBING GRAPHS
Note that at a local maximum the function changes from increasing (increasing) to decreasing (decreasing), and, similarly, at a local minimum, the function changes from decreasing (decreasing) to increasing (increasing). The absolute maximum of a function, on the other hand, is the largest value the function takes over its domain. There is an endpoint minimum at (2, 3) (denoted by the closed endpoint.) Here, the local maximum at (−1,7) is also the absolute maximum of the function.
Also indicate where the function is concave up, concave down, and identify any inflection points. Similarly, if the function decreases and then increases, a local minimum occurs at the change point.
FIRST AND SECOND DERIVATIVES
It is also the slope of the tangent to the curve at that point. Graphically, it is the rate of change of the slope of the tangent line. The slopes of the tangent lines, being first derivatives, their rate of change is the second derivative (the rate of change of the first derivative).
Here, at the maximum, the rate of change of the slope of the tangent is negative. It is correct to associate the second derivative with a change in the slope of the tangent.
CURVE SKETCHING
Identify intervals where the function increases, decreases, and has local extrema
Polynomials of degree n have at most n − 1 “turns,” are differentiable (smooth and unbroken), and tend to infinity at extremes
- APPLICATIONS OF MAXIMA AND MINIMA
- MARGINAL ANALYSIS
Alternatively, the second derivative −2k also indicates a maximum at the critical value of the first derivative, x=L∕4.). What is the largest area of a right triangle where the sum of the lengths of the shorter sides is 10 cm. What is the maximum area of a rectangular garden that can be fenced for $120 if fencing on three sides of the garden costs $5 per linear foot and on the fourth side for $7 per linear foot.
Suppose in Example 3.4.8 the circular points are cut from a large sheet which is divided into squares of side 2r. It is the study of the effect of unit monetary or product changes: "changes on the margin." The inverse relationship of supply and demand with price is a basic example.
SUPPLEMENTARY EXERCISES
EXPONENTIAL FUNCTIONS
When the bases are the same, exponents are added to quotient(28x)1∕2. c) First add numerator exponents to give 56x+4. Likewise, if exponents are equal, their bases are equated. a) First, equal bases since the variable is an exponent. The domain of y=ex is the real numbers, the range is the positive real numbers, the x-axis is an asymptote, and the de-intercept is unity.
LOGARITHMIC FUNCTIONS
As you know from the news, on the Richter scale, an earthquake with, say, a magnitude of 5 is ten times greater than a magnitude of 4, and so on. The December 2004 Indian Ocean earthquake of magnitude 9.0 on the Richter scale triggered tsunamis that caused massive destruction. The Great Chile earthquake of 1960, with a magnitude of 9.5 on the Richter scale, is the strongest earthquake ever recorded.
LOGARITHMIC FUNCTIONS 117 Logarithms and exponentials, which are inverses, sometimes make it possible to solve exponential equations, as in the following examples. Another earthquake in the Midwest, the Halloween earthquake of 1895, is estimated at 6.8 on the Richter scale.
DERIVATIVES OF EXPONENTIAL FUNCTIONS
DERIVATIVES OF NATURAL LOGARITHMS
The derivative of ln(k) is zero (it is a constant), while the derivative of ln x is 1/x.
MODELS OF EXPONENTIAL GROWTH AND DECAY
There are many applications in the biological sciences such as the prediction of fish populations and carbon dating. The half-life is the time required for an element's radioactive intensity to decrease to half of its initial value. Tritium, a radioactive isotope of hydrogen, composed of one proton and two neutrons, has a half-life of about 12.5 years.
The prehistoric art of the Lascaux caves in France is one of the famous revelations attributed to carbon-14 dating. If 1000 g of the isotope is present initially, how many grams will be there in 5 days.
APPLICATIONS TO FINANCE
For small time changes, the derivative is an estimate of the variable's rate of change. The percentage change in GDP between reporting periods is a simple representation of a growth rate. The relationship between the relative rate of change of demand and the relative rate of change of price is the elasticity of demand.
When demand is inelastic, the change in demand is in the same direction as the change in price. In exercises 10–15, determine the percent change of the functions at the indicated point.
SUPPLEMENTARY EXERCISES 1. Simplify the following expressions
PRODUCT AND QUOTIENT RULES
In words, to differentiate the product, multiply the first function by the derivative of the second and add the product of the second and the derivative of the first. In meter, "The derivative of the second times the first plus the derivative of the first times the second." The desired numerator for the times rule is the denominator (“down”) times the derivative of the numerator (“up”) minus the numerator (“up”) times the derivative of the denominator (“down”).
One mind put it in musical meter: "The upper derivative multiplied by the bottom minus the last very high derivative - all divided by the square of the lower." In Exercises 35–39, find the equation of the line tangent to f(x) at the point shown. 2x−1) where the tangent is horizontal.
THE CHAIN RULE AND GENERAL POWER RULE
- IMPLICIT DIFFERENTIATION AND RELATED RATES
- FINITE DIFFERENCES AND ANTIDIFFERENCES
The top of a ladder resting against a wall falls at one rate while the foot of the ladder moves away from the wall at a different rate. If the bottom of the ladder slides down the wall at a speed of 3 ft/s, how fast is the ladder sliding down the wall when the top of the ladder is 15 ft off the ground. Let x be the distance of the ladder from the base of the wall and y its height on the wall.
If the length of the ladder is fixed, the variables x and y are geometrically related by a right triangle axis. If the bottom of the ladder slides horizontally down the wall at 2 ft/s, how fast is the top of the ladder sliding down the wall when it is 12 ft above the ground.
SUPPLEMENTARY EXERCISES In Exercises 1–12, differentiate the given functionsc
INDEFINITE INTEGRALS
When finding the antiderivative of 3x2, the possibility of a constant in the original function follows. The function f(x) is called the integrand, C the constant of integration, F(x) the result of integration, and dx the element of the variable of integration. It is easy to check that the resulting integration is correct: its derivative must match the integrand.
To help integrate expressions with radicals or a variable in the denominator, rewrite them with fractional or negative exponents to differentiate, as in the following example. Indefinite integration leads to a family of functions that differ only by a constant called the constant of integration.
RIEMANN SUMS
One can use left endpoints, right endpoints or midpoints of the approximate rectangles to estimate areas. The formula that follows indicates how to determine the area under the curve once a choice has been made whether diexi corresponds to left endpoints, right endpoints, or midpoints. Use Riemann sums in Exercises 13−24 to approximate areas under the graph of f(x) on the given intervals.
In Problems 25–30, use the Riemann sum to approximate the areas under the graph on the given interval. Use Riemann sums with n=6 and left endpoints, right endpoints, and midpoints to approximate the marked region.
INTEGRAL CALCULUS – THE FUNDAMENTAL THEOREM
In Exercises 25−30, use a Riemann Sum to approximate areas under the graph of the given interval. b) from a Riemann Sum with n=4 and using right endpoints. The reason is that f(x) is increasing on [1, 4] and using the left endpoints underestimates the actual area.
Properties of Definite Integrals
AREA BETWEEN INTERSECTING CURVES
