Question

Consider the function f(x)=1/2(x+1)^2-3. Use the preceding/following interval method to estimate the instantaneous rate of change at 𝑥 = 1.

136

likes
678 views

Answer to a math question Consider the function f(x)=1/2(x+1)^2-3. Use the preceding/following interval method to estimate the instantaneous rate of change at 𝑥 = 1.

Expert avatar
Frederik
4.6
59 Answers
To estimate the instantaneous rate of change at 𝑥 = 1, we can use the preceding/following interval method.

The instantaneous rate of change is given by the derivative of the function. Let's find the derivative of the function f(x):

f(x)=(1/2)(x+1)^2-3

Using the power rule for differentiation, we can find the derivative:

f'(x) = 2\cdot (1/2)(x+1)^{2-1} = (x+1)

Now, to estimate the instantaneous rate of change at 𝑥 = 1, we will find the average rate of change of the function on either side of 𝑥 = 1.

Let's find the average rate of change on the interval (0, 1):

Average \ rate \ of \ change = \frac{f(1) - f(0)}{1 - 0}

Substituting the values into the equation:

Average \ rate \ of \ change = \frac{(1/2)(1+1)^2-3 - [(1/2)(0+1)^2-3]}{1}

Simplifying the equation:

Average \ rate \ of \ change = \frac{(1/2)(2)^2-3 - (1/2)(1)^2-3}{1}

Average \ rate \ of \ change = \frac{(1/2)(4)-3 - (1/2)(1)-3}{1}

Average \ rate \ of \ change = \frac{2-3 - 1/2-3}{1}

Average \ rate \ of \ change = \frac{-1/2-7/2}{1}

Average \ rate \ of \ change = \frac{-8}{2}

Average \ rate \ of \ change = -4

Similarly, let's find the average rate of change on the interval (1, 2):

Average \ rate \ of \ change = \frac{f(2) - f(1)}{2 - 1}

Substituting the values into the equation:

Average \ rate \ of \ change = \frac{(1/2)(2+1)^2-3 - [(1/2)(1+1)^2-3]}{2-1}

Simplifying the equation:

Average \ rate \ of \ change = \frac{(1/2)(3)^2-3 - (1/2)(2)^2-3}{1}

Average \ rate \ of \ change = \frac{(1/2)(9)-3 - (1/2)(4)-3}{1}

Average \ rate \ of \ change = \frac{9/2-3 - 2-3}{1}

Average \ rate \ of \ change = \frac{9/2-6 - 5}{1}

Average \ rate \ of \ change = \frac{9/2-12/2 - 5}{1}

Average \ rate \ of \ change = \frac{-3/2-5}{1}

Average \ rate \ of \ change = \frac{-13/2}{1}

Average \ rate \ of \ change = -\frac{13}{2}

Therefore, the estimated instantaneous rate of change at 𝑥 = 1 is -4 on the interval (0, 1) and -13/2 on the interval (1, 2).

Answer: The estimated instantaneous rate of change at 𝑥 = 1 is -4 and -13/2 on the intervals (0, 1) and (1, 2), respectively.

Frequently asked questions (FAQs)
Math question: Find the slope-intercept equation of a line passing through the points (2, 3) and (5, 9).
+
Math question: Graph the inequality y ≥ 2x - 3 on a coordinate plane. (
+
What is the probability of getting a head when flipping a fair coin once?
+
New questions in Mathematics
I want to divide R$ 2200.00 between Antônio, Beto and Cássia, so that Beto receives half from Antônio and Cássia receives a third of Beto. Under these conditions, how much more will Beto receive than Cássia?
5 . {2/5 + [ (8/-9) - (1/-7) + (-2/5) ] ÷ (2/-5)} . 8/15
If you have a bag with 18 white balls and 2 black balls. What is the probability of drawing a white ball? And extracting a black one?
-6n+5=-13
-6(3x-4)=-6
Equivalent expression of the sequence (3n-4)-(n-2)
Find all real numbers x that satisfy the equation \sqrt{x^2-2}=\sqrt{3-x}
sin 30
6-35 A recent study by an environmental watchdog determined that the amount of contaminants in Minnesota lakes (in parts per million) it has a normal distribution with a mean of 64 ppm and variance of 17.6. Assume that 35 lakes are randomly selected and sampled. Find the probability that the sample average of the amount of contaminants is a) Greater than 72 ppm. b) Between 64 and 72 ppm. c) Exactly 64 ppm. d) Greater than 94 ppm.
3/9*4/8=
From 1975 through 2020 the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 34 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 800? Assume the standard deviation is 1539
X~N(2.6,1.44). find the P(X<3.1)
List five numbers that belong to the 5 (mod 6) numbers. Alternate phrasing, list five numbers that satisfy equation x = 5 (mod 6)
During a month's time, an automobile sales person receives a 6% commission on the first $5000 in sales, a 7% commission on the next $5000 sales, 8% commission on anything over $10,000. What is her commission for $36,000 in sales?
find missing measure for triangle area = 48 m square base = 10m heaighy = ? m
A buyer purchased a North Carolina home for $475,250. The seller allowed the buyer to assume his first small mortgage with a loan balance of $110,000. How much is the excise tax paid in the transaction? $951 $729.50 $950.50 $221 none of the above
Pablo has a balance of $440,000 and 2/5 of the money is used to pay bills. How much money do you have left after paying the bills?
How do you convert a fraction to a decimal
3(x-4)=156
Dano forgot his computer password. The password was four characters long. Dano remembered only three characters: 3, g, N. The last character was one of the numbers 3, 5, 7, 9. How many possible expansions are there for Dano's password?