Negative numbers and coordinate plane #1 - Questions and Answers

Question 1 An ordered pair set is referred to as a ____? Answer 1 A set of ordered pairs is called a relation. If each ordered pair's x-coordinates are distinct, meaning each input of the set corresponds to just one output, then the set is more precisely referred to as a function. Answer 2 An ordered pair is a set of two elements, where the order of the elements matters. For example, the ordered pair (3,4) is not the same as the ordered pair (4,3). The first element in the ordered pair is usually denoted by 𝑥 x, and the second element by 𝑦 y. In general, an ordered 𝑛 n-tuple is a set of 𝑛 n elements, where the order of the elements matters. The elements of an ordered 𝑛 n-tuple are denoted by 𝑎 1 , 𝑎 2 , . . . , 𝑎 𝑛 a 1 ​ ,a 2 ​ ,...,a n ​ . Tuples are a fundamental concept in mathematics and computer science and are used to represent ordered collections of data. They are also used in set theory, graph theory, and other branches of mathematics. Answer 3 Explanation: An ordered pair set is referred to as a tuple. In mathematics, an ordered pair set is a set of elements in which the order of the elements matters. For example, the ordered pair set (3,4) is not the same as the ordered pair set (4,3). A tuple is a generalization of an ordered pair set to a set of any number of elements. Tuples are typically denoted by enclosing the elements in parentheses and separating them by commas, like (1,2,3) or (a,b,c). The 𝑛 n-tuple is defined as an ordered set of 𝑛 n elements. For instance, a 3-tuple could be written as ( 𝑥 1 , 𝑥 2 , 𝑥 3 ) (x 1 ​ ,x 2 ​ ,x 3 ​ ). In computer science, tuples are often used to represent a collection of related data, such as a person’s name, age, and address. In programming languages like Python, tuples are immutable and can be used as keys in dictionaries, making them a useful data structure. Question 2 Find an equation of the plane that is tangent to the given surface at the specified point 𝑧 = 3 𝑦 2 − 2 𝑥 2 + 𝑥 , ( 2 , − 1 , − 3 ) z=3y 2 −2x 2 +x,(2,−1,−3) Answer 1 Solve for the partial derivative of 𝑓 f with respect to 𝑥 x: 𝑓 𝑥 ( 𝑥 , 𝑦 ) = − 4 𝑥 + 1 f x ​ (x,y)=−4x+1 𝑓 𝑦 ( 𝑥 , 𝑦 ) = 6 𝑦 f y ​ (x,y)=6y At the point (2, -1) evaluate: 𝑓 𝑥 ( 2 , − 1 ) = − 7 f x ​ (2,−1)=−7 𝑓 𝑦 ( 2 , − 1 ) = − 6 f y ​ (2,−1)=−6 Frame the equation of the tangent plane: 𝑧 + 3 = 𝑓 𝑥 ( 2 , − 1 ) ( 𝑥 − 2 ) + 𝑓 𝑦 ( 2 , − 1 ) ( 𝑦 + 1 ) z+3=f x ​ (2,−1)(x−2)+f y ​ (2,−1)(y+1) Plug in respective values: 𝑧 + 3 = − 7 𝑥 + 14 − 6 𝑦 − 6 z+3=−7x+14−6y−6 Simplify: − 7 𝑥 − 6 𝑦 + 5 = 𝑧 −7x−6y+5=z Answer 2 You can use the partial derivatives to find a normal vector to the surface. At the point (2,-1,-3) you have 𝑓 𝑥 ( 2 , − 1 ) = − 7 f x ​ (2,−1)=−7 and 𝑓 𝑦 ( 2 , − 1 ) = − 6 f y ​ (2,−1)=−6, so that one normal vector is ( − 7 , − 6 , 1 ) (−7,−6,1). The equation of the tangent plane is given by − 7 ( 𝑥 − 2 ) − 6 ( 𝑦 + 1 ) + ( 𝑧 + 3 ) = 0 −7(x−2)−6(y+1)+(z+3)=0 which simplifies to 𝑧 = − 7 𝑥 − 6 𝑦 + 5 z=−7x−6y+5. Answer 3 To solve the equation 𝑧 = 3 𝑦 2 − 2 𝑥 2 + 𝑥 z=3y 2 −2x 2 +x for the point (2, -1, -3), we can substitute 𝑥 = 2 x=2 and 𝑦 = − 1 y=−1 into the equation and simplify: 𝑧 = 3 ( − 1 ) 2 − 2 ( 2 ) 2 + 2 z=3(−1) 2 −2(2) 2 +2 𝑧 = 3 − 8 + 2 z=3−8+2 𝑧 = − 9. z=−9. Therefore, the solution is 𝑧 = − 9 z=−9 when 𝑥 = 2 x=2 and 𝑦 = − 1 y=−1. To write the answer in the requested format, we can rewrite the equation as: 𝑧 = 3 𝑦 2 − 2 𝑥 2 + 𝑥 z=3y 2 −2x 2 +x = 3 ( 𝑦 + 1 ) 2 − 2 ( 𝑥 − 2 ) 2 + 5 =3(y+1) 2 −2(x−2) 2 +5 Answer 4 Starting with the given equation: 𝑧 = 3 𝑦 2 − 2 𝑥 2 + 𝑥 z=3y 2 −2x 2 +x We can substitute the values for 𝑥 x and 𝑦 y from the given point (2, -1, -3) to get: 𝑧 = 3 ( − 1 ) 2 − 2 ( 2 ) 2 + 2 = − 7 z=3(−1) 2 −2(2) 2 +2=−7 So the point (2, -1, -3) satisfies the equation 𝑧 = − 7 z=−7. Question 3 Water is pumped from a lower reservoir to a higher reservoir by a pump that provides 20 kW of shaft power. The free surface of the upper reservoir is 45 m higher than that of the lower reservoir. If the flow rate of water is measured to be 0.03 𝑚 3 / 𝑠 m 3 /s, determine mechanical power that is converted to thermal energy during this process due to frictional effects. Answer 1 We must first determine the pace at which the mechanical energy of the water is increasing: Δ 𝐸 mech in = 𝑚 𝑔 ℎ ΔE mech in ​ =mgh = 𝜌 𝑄 𝑔 ℎ = ( 1000   kg/m 3 ) ( 0.03   m 3 / s ) ( 9.81   m/s 2 ) ( 45   m ) = 13.23   kW =ρQgh=(1000kg/m 3 )(0.03m 3 /s)(9.81m/s 2 )(45m)=13.23kW Question 4 Find the points on the cone 𝑧 2 = 𝑥 2 + 𝑦 2 z 2 =x 2 +y 2 that are closest to the point (4,2,0). Answer Using the distance formula, the distance between a general point ( 𝑥 , 𝑦 , 𝑧 ) (x,y,z) and (4,2,0) is given by: 𝐷 = ( 𝑥 − 4 ) 2 + ( 𝑦 − 2 ) 2 + 𝑧 2 D= (x−4) 2 +(y−2) 2 +z 2 ​ Since the points are on the cone, we can replace 𝑧 2 z 2 with 𝑥 2 + 𝑦 2 x 2 +y 2 . 𝐷 = ( 𝑥 − 4 ) 2 + ( 𝑦 − 2 ) 2 + 𝑥 2 + 𝑦 2 D= (x−4) 2 +(y−2) 2 +x 2 +y 2 ​ Question 5 Find the length of the curve. 𝑟 ( 𝑡 ) = 8 𝑖 + 𝑡 2 𝑗 + 𝑡 3 𝑘 ,   0 ≤ 𝑡 ≤ 1 r(t)=8i+t 2 j+t 3 k,0≤t≤1 Answer The curve 𝑟 ( 𝑡 ) r(t) is given. To find the arc length: Arc length = ∫ 𝑎 𝑏 ∥ 𝑟 ′ ( 𝑡 ) ∥ 𝑑 𝑡 Arc length=∫ a b ​ ∥r ′ (t)∥dt where 𝑟 ′ ( 𝑡 ) = 8 𝑖 + 2 𝑡 𝑗 + 3 𝑡 2 𝑘 r ′ (t)=8i+2tj+3t 2 k.