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Mathematics

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The amount of garbage, G, produced by a city with population p is given by the function G(p). G is measured in tons per week, and p is measured in thousands of people. (a) The town of Belmont has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function G. (b) Explain the meaning of the statement G(5) = 2 . For which of the following graphs is y a function of x ? (This means x is the input and y is the output). a b c Use the graph shown below to answer the followi

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Math 248A/48A Group Work #4 – Functions and Their Graphs (Stewart 2.1/2.2) Names: ____________________________ ____________________________ _____________________________ The amount of garbage, G, produced by a city with population p is given by the function G( p) . G is measured in tons per week, and p is measured in thousands of people. (a) The town of Belmont has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function G. (b) Explain the meaning of the statement G(5) = 2 . For which of the following graphs is y a function of x ? (This means x is the input and y is the output). a b c Use the graph shown below to answer the following questions: (a) What are the coordinates of points L and K? t r a b f(x) L c x (b) Evaluate f (c) . (c) Solve f (x) = p . K  p (d) Suppose f (b) = z , then evaluate f (z) .    Math 248A/48A Group Work #4 – Functions and Their Graphs (Stewart 2.1/2.2) Use the graph shown below to find each value. Each answer will be an integer from 1 to 26. Enter answers in the second row of the table provided; leave THIRD row of the table BLANK for now. a b c d f (25) + f (26) a. f (13) e. f (1 + 4) 1  4  − f (1) + 4 4  f (1)  b. e f g h i j c. 2 f (22) f . x when k l m d. f (3) + 11 f (3 + 1) k. f (9) − f (25) f (x + 1) = 26 g. 3  f (21) + f (14) + 4 h. x when 2 f (x + 3) = 52 i. x when f (2x) = 4 l. f ( f (5) − f (1)) j. f ( f (2) + f (3)) m. f (4  6) + f (4  4) Finally, using the correspondence A = 1, B = 2, …, Z = 26, fill in letters corresponding to each number in row 2 of your table in row 3, and discover the name of a famous mathematician. Math 248A/48A Group Work #5 Name: ___________________________ ____________________________ ____________________________ Add, Subtract and Multiply Polynomials (Stewart 1.3) Perform the Indicated operation and simplify your answer. Show clear, organized work. 4(𝑥 2 − 3𝑥 + 5) − 3(𝑥 2 − 3𝑥 + 1) 2(2 − 5𝑡) + 𝑡 2 (𝑡 − 1) − (𝑡 4 − 1) (3𝑥 − 1)(2𝑥 + 1) 5(𝑥 − 3)2 Functions (Stewart 2.1) Consider the function 𝑔(𝑥) = 2𝑥 − 6 Evaluate 𝑔(9) Solve 𝑔(𝑥) = 10 Consider the function 𝑓(𝑥) = 5𝑥 2 − 𝑥 Evaluate 𝑓(−3) Simplify 𝑓(2 + ℎ) Math 48A/248A Group Work #6 – Functions and Their Graphs (Stewart 2.1/2.2) Name: _________________________ _________________________ _________________________ 1. The volume of a sphere, 𝑉 𝑟 , is a function if its radius 𝑟 given by 𝑉 𝑟 (a) Complete the table below. Give decimal approximations of the volumes rounded to one decimal place. 4 𝜋𝑟 3 (b) Find the net change in volume 𝑉 as 𝑟 changes from 1 cm to 3 cm. 𝑟 in cm 𝑉 𝑟 in cm3 1 3 5 (c) Find the net change in volume 𝑉 as 𝑟 changes from 5 cm to 7 cm. 7 (d) Compare your answers from part (b) and part (c). What does their relative size tell you about how the volume of the sphere is changing? Skip (e) Solve the equation 𝑉 𝑟 33. Round your answer to one decimal place. Math 48A/248A Group Work #6 – Functions and Their Graphs (Stewart 2.1/2.2) 2. Ken walks toward and then away from a wall. Is the graph of his distance from the wall a function of elapsed time? Why or why not? Answer the question using the definition of a function. Make a sketch of the graph. Label the axes. 3. Sketch the graph of a function that has each of the following: y f (x) has a domain of all real numbers and range f (x) 0 y g(x) has a domain x > 0 and a range of all real numbers y h(x) has a domain of all real numbers and a range of {3} 𝑦 𝑘 𝑥 has domain 𝑥 ∈ 𝑦 ∈ 1, 1 ∞, ∞ and range 4. Give the formula of a function that has each of the following domains. There are many possible answers. You can just pick one. (a) Domain 𝑥 ∈ ∞, ∞ (c) Domain 𝑥 𝜖 5, ∞ (b) Domain 𝑥 𝜖 ∞, 3 ∪ 3, ∞ ...
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