Before we can graph an inequality, we have to solve it. To help us in doing the graphing, we want to write the solution in a form of an inequality where only the variable is on the left side, and only a number is on the right side.
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In this lesson we’ll look at how to find the area of a triangle, which is equivalent to half of the product of the base and the height, A=(1/2)bh. The area is always in units of length^2 (“length squared”).
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Remember that “bi” means two, so a binomial variable is a variable that can take on exactly two values. A coin is the most obvious example of a binomial variable because flipping the coin can only result in two values: heads or tails.
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We can tell by now that these derivative rules are very often used together. We’ve seen power rule used together with both product rule and quotient rule, and we’ve seen chain rule used with power rule. In this lesson, we want to focus on using chain rule with product rule. But these chain rule/product rule problems are going to require power rule, too.
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To find the points of intersection of two polar curves, 1) solve both curves for r, 2) set the two curves equal to each other, and 3) solve for theta. Using these steps, we might get more intersection points than actually exist, or fewer intersection points than actually exist. To verify that we’ve found all of the intersection points, and only real intersection points, we graph our curves and visually confirm the intersection points.
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The reciprocal of a fraction is what you get when you turn the fraction upside down. We first saw the reciprocal when we learned about dividing by fractions, because that fraction division process required us to multiply by the reciprocal. In other words, what you get when you switch its numerator with its denominator. So the reciprocal of 3/4 is 4/3.
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At every point along a function, the function has a slope that we can calculate. If our function is a straight line, it’ll have the same slope at every point. But for any function that isn’t a straight line, the slope of the function will change as the value of the function changes. To find the slope of a function at a particular point, we can take the derivative of the function, and then evaluate it at the point we’re interested in. Doing that gives us the slope of the function at the point, but also the slope of the tangent line to the function at that point.
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Shifting the data set by a constant k means adding k to every value in the data set, or subtracting k from every value in the data set. On the other hand, scaling the data set by a constant k means multiplying or dividing every value in the data set by k.
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The perimeter of a rectangle is the length of the boundary around the figure. You can find the perimeter of a rectangle by calculating the sum of the length of its four sides. Sometimes you’ll see the perimeter formula as P=2l+2w, where l is the length of the rectangle, and w is the width of the rectangle.
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We solve inequalities the same way we solve equations, except that when we multiply or divide both sides of the inequality by a negative number, we have to do something special to it. Anytime you multiply or divide both sides of the inequality, you must “flip” or change the direction of the inequality sign.
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The simplest kind of information we’ll work with in this course is a set of individuals with one or more properties, called variables. The individuals are the items in the data set and can be cases, things, people, etc. When we construct a table, we want to think about whether we have more individuals or more variables. We’ll usually put whichever we have more of down the side of the table, so that the table is taller rather than wider.
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When you need to solve equations with variables on both sides of the equals sign, make sure to move all the variables to one side of the equation together. A simple saying that may help you remember this is “get all your x’s to Texas.” In other words, you need to move all the x terms so that they’re on the same side of the equation.
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In this lesson we’ll look at basic geometric figures like points, lines, line segments, rays, and angles, and we’ll talk about how to name them. There are very often multiple ways to name the same geometric figure. Angles, especially, are named in many different ways, so we have to be careful that we can recognize each of them.
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There are two cases to think of when you’re simplifying powers of negative bases. The first is when the base actually isn’t negative at all, because there are no parentheses around the negative sign. In that case, we’ll apply the exponent to the positive base, and then apply the negative sign afterwards. The second is when we have parentheses around the negative sign, in which case the exponent applies to the base and its negative sign.
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At any given point along a curve, we can find the acceleration vector ‘a’ that represents acceleration at that point. If we find the unit tangent vector T and the unit normal vector N at the same point, then the tangential component of acceleration a_T and the normal component of acceleration a_N are shown in the diagram below.
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Sometimes we’ll be asked for the radius and interval of convergence of a Maclaurin series. In order to find these things, we’ll first have to find a power series representation for the Maclaurin series, which we can do by hand, or using a table of common Maclaurin series.
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Compounding interest problems are a specific type of exponential growth problems and are commonly taught in calculus classes. Using certain formulas, we can see how an initial sum of money increases exponentially when we continuously add, or compound, the interest it earns to the original principal amount, and then the interest earns interest over time.
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If we’re given a double integral in rectangular coordinates and asked to evaluate it as a double polar integral, we’ll need to convert the function and the limits of integration from rectangular coordinates (x,y) to polar coordinates (r,theta), and then evaluate the integral. We can do this using the formulas to convert between rectangular and polar coordinates.
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Growth and decay problems are another common application of derivatives. We actually don’t need to use derivatives in order to solve these problems, but derivatives are used to build the basic growth and decay formulas, which is why we study these applications in this part of calculus.
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A parallelogram is a quadrilateral that has opposite sides that are parallel. So if opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. Parallelograms have opposite interior angles that are congruent, and the diagonals of a parallelogram bisect each other.
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