University:
North Central College
Course:
MATH 152 | Calculus II
Academic year:

2024
Views:

359

Pages:

4
Author:

nesikhiyawl0

Question 1 Find a recursive rule that models the exponential growth of y = 1800(1.11)ˣ Answer The functional form the given equation can be written as f(t) = 1800(1.11)ᵗ where f(0) = 1800 and 1.11 is the ratio of successive values of the function. That is, f(1)/f(0) = f(2)/f(1) = ... = f(t+1)/f(t) = 1.11 So the recursive rule for this function van be written as f(t + 1) = 1.11 * f(t) Question 2 Transform the given differential equation or system into an equivalent system of first- order differential equations. x" + 2x' + 26x = 82cos 4t Answer Transform the given differential equation or system into an equivalent system of first- order differential equations. x" + 2x' + 26x = 82cos 4t The second-order equation x" + 2x' + 26x = 82cos 4t is equivalent to system f(t, x, x') = 82 cos 4t - 26x - 2x' Hense the substitutions x₁ = x, x₂ = x' = x₁' yield the system x₁' = x₂ x₂' = 82cos 4t - 26x₁ - 2x₂ is a system of first-order equation. Question 3 Sketch a graph of the function. Use transformations of functions whenever possible. f(x) = -|x| Answer Step 1 First graph function y = |x| Then, to obtain graph of function f(x) = -|x|, do following transformations: -reflect with respect to x-axis On graph: Red - y = |x| Blue - f(x) = -|x| Step 2 Question 4 Sketch a graph of the function. Use transformations of functions whenever possible. f(x) = ⅓(x - 5), 2 ≤ x ≤ 8 Answer We need to graph the given function f(x) = ⅓|x - 5| for the restricted domain of 2 ≤ x ≤ 8. Since f(x) is linear, we can graph the function by finding its endpoints at x = 2 and x = 8 and then connecting the two points with a segment. Evaluating f(x) at x = 2 and x = 8 gives: f(2) = ⅓|2 - 5| = ⅓|-3| = -1 f(8) = ⅓|8 - 5| = ⅓|3| = 1 To graph f(x), you can then plot the endpoints of (2,-1) and (8,1) and then connect them with a segment: Question 5 The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are. x | -2 -1 0 1 2 f(x) | 31.5 10.5 3.5 10.5 31.5 g(x) | 0.2 0.6 1.8 5.4 10.8 Answer Consider the table of f(x) x | -2 -1 0 1 2 f(x) | 31.5 10.5 3.5 10.5 31.5 g(x) | 0.2 0.6 1.8 5.4 10.8 The difference of f(x) is given as 10. 5 - 31.5 = -20 2. 5 - 10.5 = -7 3. 5 - 3.5 = 7 4. 5 - 10.5 = 20 Thus, the function f(x) is not growing exponentially and thus exponential function for f(x) doesn't exist. Hence f(x)=DNE Consider the table of g(x) x | -2 -1 0 1 2 g(x) | 0.2 0.6 1.8 5.4 10.8 as we can that g(x) has same multiplier as 3 common ratio 0. 6 / 0.2 = 3 8 / 0.6 = 3 4 / 1.8 = 3 8 / 5.4 = 3 Hence common ratio b=3 and initial value is a=1.8 Hence, g(x) = 1.8(3)ˣ