Below is the graph of f '(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1 and x = 2 and a relative maximum at x = -1.5 and a relative minimum at x = 1.5. Which of the following statement is true?
(it's a positive cubic function graphed with positive maximum at x = -1.5 and a negative minimum at x = 1.5)
f is concave up from x = 0 to x = 3.
f has an inflection point at x = 0.
f is concave down from x = -1.5 to x = 1.5.
None of these is true.
Which of the following functions grows the fastest as x grows without bound? (5 points)
f(x) = e^x
g(x) = ecosx
h(x) = (5/2) ^x
They all grow at the same rate.
Use Euler's Method with two equal step sizes to estimate the value of y(0.4) for the differential equation y ' = x + y, with y(0) = 1.
If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.48). (5 points)
since f"(x) = 0 at x = -1.5 and x = 1.5, and f"(x) < 0 in (-1.5,1.5), C is true
exponentials grow fastest, and the larger the base, the faster they grow. So, what do you think?
I'll let you apply Euler's method. It just involves extrapolating along the tangent line.
okay thanks for the first one!
and since 5/2 > e the answer should be C.
and the third one is 1.48?
oops I meant 5/2 < e so the answer would be f(x) not h(x)
To determine which statement is true about the graph of the derivative function f '(x), we can use the information given. The x-intercepts of f '(x) are at x = -3, x = 1, and x = 2. This means that f(x) has relative maximum and minimum points at the corresponding x-values.
Since the function is a positive cubic function with a positive maximum at x = -1.5 and a negative minimum at x = 1.5, we can conclude that the statement "f is concave down from x = -1.5 to x = 1.5" is true. This is because a positive cubic function has a concave down shape when it has a negative minimum and positive maximum.
Therefore, the correct answer is: f is concave down from x = -1.5 to x = 1.5.
For the second question regarding which function grows the fastest as x grows without bound, we can compare the growth rates of the given functions.
The function f(x) = e^x grows the fastest as x grows without bound. This is because the exponential function e^x has an infinite growth rate. The function g(x) = ecosx is a periodic function that oscillates between -1 and 1 as x increases, so its growth rate is limited. The function h(x) = (5/2)^x is an exponential function, but its base is less than e, so it grows at a slower rate than f(x)=e^x.
Therefore, the correct answer is: f(x) = e^x.
To use Euler's Method to estimate the value of y(0.4) for the differential equation y ' = x + y, with y(0) = 1, we need to take equal step sizes and iterate through the process.
Step 1: Set up the initial values:
x0 = 0
y0 = 1
Step 2: Determine the step size. Since we need to use two equal step sizes, we can divide the interval [0, 0.4] into two parts: [0, 0.2] and [0.2, 0.4].
Step size, h = (0.4 - 0) / 2 = 0.2
Step 3: Use Euler's Method to iterate through the process.
For the first step, i = 1:
xi = x0 + i * h = 0 + 1 * 0.2 = 0.2
yi = y0 + h * (x0 + y0) = 1 + 0.2 * (0 + 1) = 1 + 0.2 = 1.2
For the second step, i = 2:
xi = x0 + i * h = 0 + 2 * 0.2 = 0 + 0.4 = 0.4
yi = y0 + h * (x0 + y0) = 1 + 0.2 * (0 + 1.2) = 1 + 0.24 = 1.24
Therefore, the estimated value of y(0.4) using Euler's Method with two equal step sizes is 1.24.
Hope this explanation helps! Let me know if you have any further questions.