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# Calculus 1 Week 1 Review Notes MAT-271

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This 9 page Class Notes was uploaded by Emma Drum on Thursday August 18, 2016. The Class Notes belongs to MAT-271 at Caldwell Community College and Technical Institute taught by Michelle Powell in Fall 2016. Since its upload, it has received 11 views. For similar materials see Calculus I in Math at Caldwell Community College and Technical Institute.

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Date Created: 08/18/16

Review Notes for Calculus I Symmetry: A graph is symmetric with respect to the y-axis if whenever (x, y) is a point on the graph then (-x, y) is also a point on the graph. Some even functions (y=x , y=x , etc.) have symmetry with respect to the y-axis. These graphs usually are parabolas (u-shaped graphs). To figure out if a graph has y-axis symmetry, then you replace all x’s with the opposite of x (ex: x would become –x). A graph is symmetric with respect to the x-axis if whenever (x, y) is a point on the graph then (x, -y) is also a point on the graph. These graphs can look like a parabola turned on its side. To figure out if a graph has x-axis symmetry, you replace all y’s with the opposite of y (ex: y would become –y). A graph is symmetric with respect to the origin if whenever (x, y) is a point on the graph the (-x, -y) is also a point on the graph. Some odd functions (y=x ,y=x , etc.) have symmetry with respect to the origin. To figure out if a graph has origin symmetry, you replace all x’s with the opposite of x and all y’s with the opposite of y (ex: x would become –x and y would become –y). Note: After replacing the variables with the opposite of that variable, and after simplifying the equation, if the resulting equation is not the same as the equation you started with, then there is no symmetry. Ex: y=2x -x Y-axis 3 y=2x -x y=2(-x) -(-x) (Replace x with –x) y=-2x +x (Do the exponent first, then multiply it by 2. Since the exponent is negative, it stays negative. Then simplify the other x.) It is not symmetric with respect to the y-axis. We know this because the final equation is not the same as the equation you started with. X-axis y=2x -x -y=2x -x (Replace y with –y.) There is actually nothing else you can do here. The only thing you could do is divide everything by -1, which would make the other side negative as well, resulting in an equation that is different than the initial equation. It is not symmetric with respect to the x-axis. Origin y=2x -x 3 -y=2(-x) -(-x) (Replace x with –x and y with –y) -y=-2x +x (Simplify the equation) y=2x -x (Divide by -1) It is symmetric with respect to the origin. Here, you can divide by -1 to get back to the original equation because you have to replace both variables. The resulting equation is the same as the initial equation, so it is symmetric with respect to the origin. Review Notes for Calculus I: Slope: Ris e Slope is always rise over run: Ru This means that, on a n graph, the value on top is how many spaces you move up or down, and the bottom value is how spaces you move right or left. o Example: Slope= 2/3 o In this example, the slope is 2/3. To get to a new point on the line, you would go up 2 spaces and to the right 3 spaces. Slope The change equals the change in y over the change in x: in y The change in x In math, the Greek letter ∆ (delta), means “the change in.” So the formula would actually be ∆ y= y2−y1 ∆ x x 2−x 1 = slope The letters represent the numbers in the coordinates of two points: (x , y ) 1 1 and (x 2 y2). To find slope, two points have to be given. Ex: Find the slope using these two points: (-2, 0) and (3, 1) ∆ y= y2−y1 Slope= ∆ x x2−x1 (I usually rewrite the formula before starting the problem to help me remember it). 1−0 Slope= 3−(−2) (Plug in the numbers. The first coordinate is (x1, y1) and the second coordinate is (x2, y2), unless otherwise specified.) Slope= 1 5 (Subtract the top and bottom.) Note: If the fraction ends up having a zero on the bottom, the slope is undefined, and the graph will look like a vertical line. If the fraction ends up having a zero on top, the slope equals zero, and the graph will like a horizontal line. Review Notes for Calculus I: Functions: A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range o If the domain corresponds to more than one member of the range, then it is not a function. The range can correspond to more than one member of the domain and it is still a function. x y x y 1 4 1 3 2 6 2 4 4 9 2 7 NOT a function IS a function o Vertical Line Test: used to test if a graph is a function. The graph on the left does not pass the vertical line test because the graph intersects the vertical line at two different points at the same time. The graph on the right does pass because o Domain and Range: Most questions concerning functions will ask what the Domain and the Range of the function is. Domain is all the x-values that are used as inputs in the graph. In the example on the left, all of the x-values used are between 2 and 13. However, in the example on the right, the domain will continue reaching outward, so all x- values will be used. This is denoted by (-∞, ∞), meaning that the x-values continue to negative infinity and positive infinity Range is all the y-values that are used as inputs in the graph. In the example on the left, all of the y-values used are between 1 and 8. In the example on the right, the range starts at -6. It then continues upward into positive infinity. Note: You may notice that brackets and parentheses are used. There are used for to denote different things. Brackets, [], are used when a number is included. For example, the range in the example on the right, has an included -6. It is included because the range has a definite point where it cuts off. It doesn’t go below -6. Parentheses, (), are used when a number isn’t included. In the example on the right, the domain uses parenthesis because infinity doesn’t have a definite end point. Infinity never cuts off at a certain point. Parentheses are also used when a point on the graph is open. It looks like this Curly Brackets, {}, are used if you have individual points on the graph. Let’s say that there are three points on the graph (1, 2), (3, -4), and (5, 9). The domain would be {1, 3, 5} and the range would be {- 4, 2, 9}. 2 Ex: Find the domain and range of f(x)=3x +6x+1. Then find f(2) and f(b-1). Domain D: (-∞, ∞) (The easiest way to find the domain is to look at the graph. We can see that the graph will hit all value of x) Range R: [-2, ∞) (Looking at the graph is also the easiest way to find the range. The minimum of the graph is (-1, -2). So, the range starts at -2 and continues on forever.) f(x)=3x +6x+1 f(2)=3(2) +6(2)+1 (f(2) just means that you are substituting x for 2, so plug in 2 where you see an x.) f(2)=3(4)+6(2)+1 (Use order of operations: exponent first). f(2)=12+12+1 (Multiply 3 and 4, then 6 and 2) f(2)=25 (Add it all together) f(x)=3x +6x+1 f(b-1)=3(b-1) +6(b-1)+1 (Plug in b-1 wherever you see an x.) f(b-1)=3(b-1)(b-1)+6(b-1)+1 ((b-1) will expand out to (b-1)(b-1)) 2 f(b-1)=3(b -2b+1)+6(b-1)+1 (Use the FOIL method to simplify (b-1) (b-1)) f(b-1)=3b -6b+3+6b-6+1 (Distribute 3 to b -2b+1 and 6 to b-1) f(b-1)=3b +4 (Combine like terms: the -6 and 6 cancel out and 3+1=4) Review Notes for Calculus I: Writing the Equation of a line: Point-Slope Form o One way to write the equation of a line is to use point-slope form. o You are given the slope and one point on the graph o Formula: y – y =1m(x – x ) 1 m=slope x1= the x value of the point given. y1= the y value of the point given. Ex: Write the equation of a line if (1, -2) lies on the line and slope=3. y-y1= m(x-x )1 (I find rewriting the formula helps me remember it better) y+2 = 3(x-1) (Plug in the numbers. Note: the -2 becomes a positive 2 because the formula already has a minus sign there -(2) = -2 ) y+2 = 3x-3 (Distribute the 3 to the x and the -1) y = 3x-5 (Subtract 2 from both sides) Slope-Intercept Form o Another way to write the equation of a line is to use slope-intercept form. o You can be given two points on the graph or the slope and one point on the graph. o Formula: y=mx+b m=slope b=y-intercept x=the input value y=the output value Ex: Write the equation of a line if these points (-2, -2) and (1, 7) are on the graph. 7−(−2) Slope= (Find the slope first using the slope formula: 1−(−2) y2−y1 Slope= ) x2−x1 9 Slope= 3 (Subtract the top and bottom) Slope=3 (Divide) y=mx+b (Now we are going to use the slope-intercept form formula) -2=3(-2) +b (Plug in the numbers. Use the first coordinate that is given. This is to find the y-intercept because it is not given.) -2=-6+b (Multiply the 3 and -2) b=4 (Add 6 to both sides) y=3x+4 (Plug the slope and the y-intercept into the formula and now you have the equation of the line!) Ex: Write the equation of a line if the point (3, -12) is on the line and slope=-2 y=mx+b (Since we know the slope this time, we don’t have to use the slope formula to find it first.) -12=-2(3) +b (Plug in the numbers. -12=-6+b (Multiply the -2 and 3) b=-6 (Add 6 to both sides) y=-2x-6 (Plug the slope and the y-intercept into the formula and now you have the equation of the line!)

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