...you some practice with algebra and with graphing. Also, this worksheet introduces the idea of “tangent lines” to circles. Later on in Math 124, you’ll learn how to find tangent lines to many other types of curves. 1. Two circles, called C1 and C2 , are graphed below. The center of C1 is at the origin, and the center of C2 is the point in the first quadrant where the line y = x intersects C1 . Suppose C1 has radius 2. C2 touches the x and y axes each in one point. What are the equations of the two circles? y y=x C2 x C1 Worksheet Math 124 Week 1 2. Let C be the circle of radius 5 centered at the origin. The tangent line to C at a point Q is the line through Q that’s perpendicular to the radial line connecting Q to the center. (See picture.) Use this information to find the equations of the tangent lines at P and Q below. y Q P x Note: Later in Math 124, you’ll learn how to find tangent lines to curves that are not circles! Page 2 Worksheet Math 124 Week 1 3. Sketch the circle of radius 2 centered at (3, −3) and the line L with equation y = 2x + 2. Find the coordinates of all the points on the circle where the tangent line is perpendicular to L. y x Page 3 Worksheet Math 124 Week 1 4. Draw the circle with equation x2 +y 2 = 25 and the points P = (−3, −4) and Q = (−8, 0). Explain why P is on the circle. Is the line through P and Q tangent to the circle? How do you know? Page 4...
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...set is Monday 17.9. at 9:45 (before the first exercise group). Please submit your exercise answers to the course box on the 2nd floor of the Chydenia building. Problem 1. (a) Consider the graph of f (x) = x2 and the line tangent to f (x) at (a, f (a)). The equation of the 1 2 tangent line is y = − 5 x − 25 . Find a, f (a), and the slope of the parabola f (x) at (a, f (a)). 6 (b) Consider the graph of y = x3 . Find the point(s) on the graph where the slope is equal to 11 . (c) The demand function for a commodity with price P is given by the formula D(P ) = a − bP . Find dD(P )/dP . (d) The cost of producing x units of a commodity is given by the formula C(x) = p + qx2 . Find C (x), the marginal cost. Problem 2. (a) Determine the limit limx→0 (3 + 2x2 ). (b) Determine the limit limx→2 (2x2 + 5)3 2 (c) Determine the limit limx→1 x +7x−8 (tip: modify the numerator using similar approach as in the x−1 lecture example) (d) Let f (x) = 4x2 . Show that f (5 + h) − f (5) = 40h + 4h2 . Hence, f (5 + h) − f (5) = 40 + 4h h Using this result, find f (5). Problem 3. Use differentiation rules (not the formal definition) to do the following: (a) Differentiate y = √ 3 − 6x2 + 49x − 54 2x 2 (b) Differentiate y = x − x 2 (c) Differentiate y = (x + 3x − 5)3 (d) Find the equation of the tangent line to the graph of f (x) = x/(x2 + 2) at x0 = 3. Tip: Given a point on a line and the slope, one can construct an equation of the line. Problem 4. Give an example of a function (if possible), which is defined for all...
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...2 85. Since A A Section 2.2 Trigonometric Functions of Non-Acute Angles 8. Answers may vary. in QIII is The reference angle for an angle given by 180 . The trigonometric functions of are as follows: sin sin csc csc cos cos sec sec tan tan cot cot 1. C; 180 98 82 (98° is in quadrant II) 2. F; 212 180 = 32° (212° is in quadrant III) 3. A; 135 360 225 and 225 180 = 45° (225° is in quadrant III) 9. Answers may vary. Two coterminal angle have the same values for their trigonometric functions because the two angles have the same reference angle. 4. B; 60 360 300 and 360 300 60 (300° is in quadrant IV) 10. Answers may vary. In quadrant II, the sine function is positive while the cosine and tangent functions are negative. 5. D; 750 2 360 30 (30° is in quadrant I) cos tan 11. 30 1 2 3 2 12. 45 2 2 13. 60 3 2 2 2 1 2 cos 120 cos 60 1 2 14. 120 3 2 15. 135 2 2 16. 150 sin150 sin 30 1 2 2 2 3 2 60 7. 2 is a good choice for r because in a 30 60 right triangle, the hypotenuse is twice the length of the shorter side (the side opposite to the 30 angle). By choosing 2, one avoids introducing a fraction (or decimal) when determining the length of the shorter side. Choosing any even positive integer for r would have this result; however, 2 is the most convenient value. 86. Yes, the third angle can be found by subtracting the given acute angle...
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...Creating a part in Pro/ENGINEER: Select the FILE menu at the top of the screen and then select the option NEW Select the button for PART (if it is not already selected) and enter a name for the file in the space provided. This will create a set of three mutuallyorthogonal datum planes and display them on the screen. They are labeled RIGHT, TOP, and FRONT A few comments on Datum Planes; Datum planes are 2-sided. They have a Yellow side and a Red side. The yellow side is considered primary, that is, in most commands involving orientation of the datum plane in 3D space, the facing of the yellow side takes precedence. To create your first shape feature: From the Feature menu select CREATE For Feature Type select SOLID and in the SolidType menu, select PROTRUSION (a protrusion is a solid feature which adds material to the model) For SolidOptions select EXTRUDE, SOLID and DONE This specifies that we will be creating our protrusion using the operation of extrusion, which sweeps a planar profile through a distance normal to the plane of the profile. The FeatureControl dialogue box will now open and show what is required to create the feature in question. The box lists the properties for the feature and notes whether they have been defined, are currently being defined, or are still required. For Attributes select One Side and Done (this option specifies that we will extrude to one side of the plane of the profile; the two side option makes the profile plane the midplane of...
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...Circles & Angles: The following series of worksheets were written to help students discover some relationships with angles that are created by tangents, chords, and secants in circles. Lesson: Central and Inscribed Angles Grade Level: Secondary Level (Geometry) Sunshine State Standard: MA.A.1.4.2, MA.A.2.4.2, MA.B.1.4.2, MA.B.2.4.1, MA.B.4.4.1, MA.C.1.4.1, MA.C.2.4.1, MA.D.1.4.1 Materials: • Students: The use of GeoGebra dynamic worksheets • Teachers: Projection of GeoGebra dynamic worksheets Objectives: 1. Students will discover properties of an angle inscribed in a circle 2. Students will discover properties of the interior angles of a cyclic quadrilateral. 3. Students will discover properties of angles that are formed when two chords of a circle intersect. 4. Students will discover properties of angles formed by two intersecting secants of a circle. 5. Students will be able to find the center of a circle. Vocabulary: tangent line (segment), secant line (segment), central angle, inscribed angle, cyclic quadrilateral, chord, diameter, radius, right angle, arc (major and minor), intercepted arc, measure (angles and arcs), center, perpendicular bisector Lesson Plan: (These lessons should be taught during a unit on circles. Each separate dynamic worksheet topic will probably take one class setting, approximately 50 minutes.) -To start a discussion about circles it may be a good idea to discuss some definition of terms that deal with a circle. Displaying a image similar to...
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...7, 11 and 12 require some Maple work. 1. Solve the following inequalities: a) b) c) 2. Appendix D #72 3. Consider the functions and . a) Use a Maple graph to estimate the largest value of at which the graphs intersect. Hand in a graph that clearly shows this intersection. b) Use Maple to help you find all solutions of the equation. 4. Consider the function. a) Find the domain of. b) Find and its domain. What is the range of? c) To check your result in b), plot and the line on the same set of axes. (Hint: To get a nice graph, choose a plotting range for bothand.) Be sure to label each curve. 5. Section 1.6 #62 6. Section 2.1 #4. In d), use Maple to plot the curve and the tangent line. Draw the secant lines by hand on your Maple graph. 7. Section 2.2 #24. Use Maple to plot the function. 8. Section 2.2 #36 9. Section 2.3 #14 10. Section 2.3 #26 11. Section 2.3 #34 12. Section 2.3 #36 Recommended Problems Appendix A all odd-numbered exercises 1-37, 47-55 Appendix B all odd-numbered exercises 21-35 Appendix D all odd-numbered exercises 23-33, 65-71 Section 1.5 #19, 21 Section 1.6 all odd-numbered exercises 15-25, 35-41, 51, 53 Section 2.1 #3, 5, 7 Section 2.2 all odd-numbered exercises 5-9, 15-25, 29-37 Section 2.3 all odd-numbered exercises...
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...All of the tangents in Beowulf add artistic elements to the story and provide historical contexts for readers so they can understand the purpose better. These kinds of stories are meant to teach and entertain people at the same time. . Beowulf’s main goal was to be a hero. Most men in this story wanted to be just like him. Additionally, he followed, what he perceived to be, the will of god. This is why he killed the dragon; because he believe that that is what God wanted him to do. Unferth showed the more jealous side of those who are heroic and successful. Additionally, the role of women in this poem are vital because they keep the men in order, but showing them how they should or should not act. The social world of Beowulf exclude non-heroics and monsters...
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...the Equation of a Tangent Line Using the First Derivative Certain problems in Calculus I call for using the first derivative to find the equation of the tangent line to a curve at a specific point. The following diagram illustrates these problems. There are certain things you must remember from College Algebra (or similar classes) when solving for the equation of a tangent line. Recall : • A Tangent Line is a line which locally touches a curve at one and only one point. • The slope-intercept formula for a line is y = mx + b, where m is the slope of the line and b is the y-intercept. • The point-slope formula for a line is y – y1 = m (x – x1). This formula uses a point on the line, denoted by (x1, y1), and the slope of the line, denoted by m, to calculate the slope-intercept formula for the line. Also, there is some information from Calculus you must use: Recall: • The first derivative is an equation for the slope of a tangent line to a curve at an indicated point. • The first derivative may be found using: A) The definition of a derivative : lim h →0 f (x + h ) − f ( x ) h B) Methods already known to you for derivation, such as: • Power Rule • Product Rule • Quotient Rule • Chain Rule (For a complete list and description of these rules see your text) With these formulas and definitions in mind you can find the equation of a tangent line. Consider the following problem: Find the equation of the line tangent to f ( x ) = x 2...
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...as (a) Chord (b) Secant (c) Tangent (d) none of these Q.2 number of tangents that can be drawn through a point which is inside the circle is (a) 3 (b) 2 (c) 1 (d) 0 Q.3 A line through point of contact and passing through centre of circle is known as (a) tangent (b) Chord (c) normal (d) segment Q.4 A circle is inscribed in a triangle with sides 3, 4 and 5 cm. The radius of the circle is (a) 6 cm (b) 5 cm (c) 4 cm (d) none of these Q.5 Distance between two parallel lines is 10 cm. The radius of circle which will touch both two lines is (a) 5cm (b) 7 cm(c ) 12 cm (d) None of these Section B. 2 Mark Each Q.6 In figure, CP and CQ are tangents to a circle with centre O. ARB is anothertangent touching the circle at R. If CP = 12 cm, and BC = 8cm, then find the length of BR. Q.7 In figure AB is a chord of the circle and AOC is its diameter such that ABC 500 . If AT is the tangent to the circle at the point A, find BAT www.cbsesmart.weebly.com “Chase Excellence- Success Will Follow” ll Follow” “Chase Excellence- Success Will Follow” 2011 For more free sample papers and test papers Visit http://jsunilclasses.weebly.com company address] Page 2 Q.8 Two tangents PA and PB are drawn to the circle with centre, such that APB 1200 . Prove that OP = 2 AP. Section c 3 Mark Each Q.11 The tangent at a point C of a circle and a...
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...his family and the school itself and follows the trail of his mysterious guest. The film mentions the universe of tangent. In fact, it clarifies the concept for the movie. We live in the real universe right now. And time continues to flow without interference. But sometimes, in accidents, there's a break in time in incidents, and an alternate universe can occur and it's called the Tangent universe. The tangent universes are extremely unstable and in a short time it turns...
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...The first step that I did to determine the values of cosine, tangent, and sine was to sketch a right triangle that has the vertical length of three and has the horizontal base length of two. The second step that I did to determine the values of cosine, tangent, and sine was to utilize the pythagorean theorem to determine the length of the hypotenuse which looks like this square root(2^2 + 3^2) = square root(13). The third step that I did to determine the values of cosine, tangent, and sine was to utilize the acronym SOH-CAH-TOA to determine the right trigonometry ratios that have the opposite side always being vertical side which is three and the adjacent side always being the horizontal side which is two. The fourth step that I did to determine...
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...the tangent universe, which he has no idea of. The happenings in the movie Donnie Darko built up a purpose for him as well as an understanding as to why he was in the tangent universe in the first place. However, the realization that he was not in the primary universe, Donnie does not have a clear purpose to serve in the tangent universe and various elements of his journey seem unclear. For example, in the tangent universe, Donnie does not seem to have met Gretchen and neither does he connect well with Frank. However, following the fact that Gretchen and Frank have a purpose to accomplish in the favor of saving the universe before it collapses into a black hole, Donnie realizes that there is instability in the tangent universe and it is his responsibility as the chosen one to hand over the jet engine artifact back to the primary universe before things escalate out of proportion. Bundled with supernatural powers such as gaining some sort of powers, such as the capabilities to conjure fire, water, and telekinesis, Donnie, firstly did not have an idea as to what these powers entailed and why he had to have them. However with provocation from other characters in the tangent universe, Donnie is able to understand that time is not in his or the primary universe’s side. The artifact, jet engine, was in tangent universe as a mistake as there can never be two copies of the same item in one universe. With this understanding, the purpose of being in the tangent universe...
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...trigonometric functions of any angle. Notice the side opposite the angle θ has a length of the y value of the given coordinates. The adjacent side has a length of the x value of the coordinates. The length of the hypotenuse is given by x2 + y2 . Lets say, for the sake of argument, the length of the hypotenuse is 1 unit. This would mean the following would be true. 1 y 1 cos θ = x sec θ = x y x tan θ = cot θ = x y You must think of the sine function as giving you the y value, whereas the cosine function yields the x value. This is how we will determine whether the sine, cosine, tangent, cosecant, secant or cotangent of a given angle is a positive or negative value. sin θ = y csc θ = If the angle to be evaluated is in quadrant IV, for instance, the sine of the angle θ will be negative. The cosine of θ , in this instance, will be positive, while the tangent of the angle θ will be negative. Example Evaluate the six trigonometric functions of an angle θ , in standard position, whose terminal side has an endpoint of (-3,2). The angle with terminal side is first drawn. Remember, in order to evaluate the six trigonometric ' functions for θ , use the reference angle θ . From the endpoint of the terminal side of the angle, a line is drawn to the x axis. This is the reason reference angles are always drawn in relation to the x axis. It will always create a right triangle with which to work. Now all that is needed to solve the problem, is...
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...4-5 Exact Values of Sine, Cosine, and Tangent RECALL: All Students Take Calculus | |Q1 |Q2 |Q3 |Q4 | |SINE (y) |+ |+ |- |- | |COSINE (x) |+ |- |- |+ | |TANGENT |+ |- |+ |- | *ALL positive *Sine positive *Tangent positive * Cosine pos ALL Students Take Calculus KEY POINTS (COS [pic], SIN [pic]) WE ALREADY KNOW: [pic]= 0/360 degrees [pic] = 90 degrees [pic] = 180 degrees [pic] = 270 degrees POINTS WE WILL MASTER TODAY: [pic] = 45 degrees [pic] = 30 degrees [pic] = 60 degrees Consider a rotation of 45 degrees ( or [pic] radians) What type of triangle do you get? (45-45-90 Right Triangle) BUT we need a radius/hypotenuse of 1!!! ((((( When [pic] = [pic], cos [pic] = [pic] sin [pic] = [pic] SYMMETRY AROUND THE UNIT CIRCLE WHEN [pic] = multiples of[pic] [pic] *Value in the first quadrant is at [pic] = [pic] *LOOK AT SECOND QUADRANT WHERE [pic] = [pic] *Numbers are...
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...cosine. Again, take the horizontal axis to be the t-axis, but now take the vertical axis to be the x-axis, and graph the equation x = cos t. Note that it looks just like the graph of sin t except it's translated to the left by /2. That's because of the identity cos t = sin (/2 + t). Although we haven't come across this identity before, it easily follows from ones that we have seen: cos t = cos –t = sin (/2 – (–t)) = sin (/2 + t). The graphs of the tangent and cotangent functions The graph of the tangent function has a vertical asymptote at x = /2. This is because the tangent approaches infinity as t approaches /2. (Actually, it approaches minus infinity as t approaches /2 from the right as you can see on the graph. You can also see that tangent has period ; there are also vertical asymptotes every units to the left and right. Algebraically, this periodicity is expressed by tan (t + ) = tan t. The graph for cotangent is very similar. This similarity is simply because the cotangent of t is the tangent of...
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