University:
Carleton College
Course:
MATH 120 | Calculus 2
Academic year:

2024
Views:

103

Pages:

3
Author:

Nevaeh Bauer

Question 1 Given the following statements: I. The function f(x) = x²sin²x is an odd function. II. The function f(x) = eˣsin x is a neither function. III. The function f(x) = cos (5x)sin²x is an even function IV. The function f(x) = (x - 5)e²ˣ is an even function. Answer 1 Step 1 we know that: put x = -x and then, f(-x) = f(x) → even function f(-x) = -f(x) → odd function f(-x) ≠ -f(x) or f(x) → neither function Step 2 I) The function f(x) = x²sin²x is odd function So, put x = -x in function: f(-x) = (-x)²(sin (-x))² = x²(-sin x)² f(-x) = x²(sin x)² So, f(-x) = f(x) [this is a even function not odd function] (statement false) Answer 2 II) The function f(x) = eˣsin x is a neither function: put x = -x f(-x) = e⁻ˣsin (-x) = e⁻ˣ(-sin x) f(-x) = -e⁻ˣsin x f(-x)≠ f(x) or -f(x) So, This is a neither function (statement true) III) Step 3 The function f(x) = cos(5x)sin²x is even function So, put x = -x f(-x) = cos(5(-x)) (sin(-x))² = cos(-5x) (-sin x)² f(-x) = cos(5x) sin² x = f(x) So, This is a even function IV) The function f(x) = (x - 5)e²ˣ is an even function. put x = -x So, f(-x) = (-x - 5)(e²(-x)) = -(x+5)e⁻²ˣ ≠ -f(x) or f(x) So, This is a neither function Question 2 If f(x) = 4x - 3, show that f(2x) = 2f(x) + 3 Answer Given that the function f(x)=4x-3 Plugging x=2x both sides we have f(2x)=4(2x)-3 =8x-3 =2(4x-3)+3 =2f(x)+3 Therefore, we get f(2x)=2f(x)+3 Question 3 For the expression: y = 4x-2, complete the following table: X: | 1 | 3 | 6 | 10 | Y: | | | | | Answer To complete the table, you first need to substitute each of the x-values into y = 4x - 2 to find the corresponding y-values. This gives: x = 1 : y = 4(1) - 2 = 4 - 2 = 2 x = 3 : y = 4(3) - 2 = 12 - 2 = 10 x = 6 : y = 4(6) - 2 = 24 - 2 = 22 x = 10 : y = 4(10) - 2 = 40 - 2 = 38 The table is then completed as: X: | 1 | 3 | 6 | 10 | Y: | 2 | 10 | 22 | 38 | Question 4 How can modifications to the parameters affect how the graphs of exponential functions are translated? Answer A function with an exponential shape is y = a(b)⁽ˣ⁻ʰ⁾ + k. h represents a horizontal translation since it affects the x-values. Graph translation to the appropriate h units occurs if h>0. For example, y = 3⁽ˣ⁻²⁾ is the parent function of y = 3ˣ translated 2 units to the right. If h < 0, the graph is translated to the left |h| units. For example, y = 3⁽ˣ⁺²⁾ is the parent function of y = 3ˣ translated | - 2 | = 2 units to the left. Given that it changes the y-values, k represents a vertical translation. If k > 0, the graph is translated up k units. For example, y = 3ˣ + 2 is the parent function of y = 3ˣ translated up 2 units. If k < 0, the graph is translated down |k| units. For example, y = 3ˣ - 2 is the parent function of y = 3ˣ translated down | - 2 | = 2 units.