Watch and learn now! Let \(s(t)\) be a function giving the position of an object at time t. A ball is dropped from a height of 64 feet. Find the maximum rate of change of the function f(x,y,z) = \frac{8x+8y}{z} at the point (6,7,-1). Could someone please explain in a step by step fashion; how to solve for Vf in the first kinematic equation: We want to get Vf = something, so we start by multiplying both sides of the equation: In example 4 when plugging in the formulas there were no step by steps on how they got the answer.What are they someone please (final velocity) squared = some formula with known values. The total area will be the sum of the areas of the blue rectangle and the red triangle. All content of site and practice tests copyright 2017 Max. What is the maximum value of a function whose derivative has no roots? Use the Second Derivative Test where applicable. While it's true that there is more gravitational force acting on a heavier object, this doesn't correspond to an increase in acceleration. This is called the Second Derivative Test. ( 5 votes) Upvote Flag \begin{align} Find the maximum slope of the curve: -x^3 + 6x^3 + 2x + 1. Assume that the number of barbeque dinners that can be sold, \(x\), can be related to the price charged, \(p\), by the equation \(p(x)=90.03x,0x300\). Let f(x,y)=x^2+y^2-x-2y. Substitute the measurements for force, distance and mass into the equation. It might seem like the fact that the kinematic formulas only work for time intervals of constant acceleration would severely limit the applicability of these formulas. Again, we used other kinematic formulas, which have a requirement of constant acceleration, so this third kinematic formula is also only true under the assumption that the acceleration is constant. Question: How to calculate maximum velocity from a derivative? We usually start with acceleration to derive the kinematic equations. It's lucky since we don't need to know the mass of the projectile when solving kinematic formulas since the freely flying object will have the same magnitude of acceleration, We choose the kinematic formula that includes, For instance, say we knew a book on the ground was kicked forward with an initial velocity of, To choose the kinematic formula that's right for your problem, figure out. Suppose f(x, y) = e \ 2x+3y , P = (1, 0) and ~v = 3~i - 4~j. I mean, what relation have between calculating distance of volacity of the fuction in the given arrange of t and using differential? Find the maximum rate of change of f at the given point and the direction in which it occurs. The height after t t seconds is: s(t) = 32 + 112t 16t2 s ( t) = 32 + 112 t 16 t 2. Find the maximum value of this derivative at this point. Find the gradient of the function at the given point. I found the derivative of this velocity . I understand that these equations are only for acceleration being constant. Find the maximum value of the directional derivative at (-2, 0), and find the vector in the direction in which the maximum value occurs. The acceleration vector of the enemy missile is. (a) Find the directional derivative of f at P in the direction of Q (-2,1,4) (b) Find the direction of the maximum rate of change o, Find the value of the derivative (if it exists) at the given extremum. Arrow between those points and enter your best guess of the position of the maximum. Evaluating these functions at \(t=1\), we obtain \(v(1)=1\) and \(a(1)=6\). Level of grammatical correctness of native German speakers. The population growth rate is the rate of change of a population and consequently can be represented by the derivative of the size of the population. If the sign of the second derivative is negative, then the curve has a local maximum there. F(x,y) = 7e^{x}\sin{y}, P=(0, \frac{\pi}{3}), \textbf{v}=<-5,12>. What is the average velocity during its fall? The actual revenue obtained from the sale of the \(101^{\text{st}}\) dinner is, \(R(101)R(100)=602.97600=2.97,\) or \($2.97.\). The kinematic formulas are a set of formulas that relate the five kinematic variables listed below. If we take the derivative of the velocity, we can find the acceleration, or the rate of change of velocity. Thus, the velocity is a minimum at t = 2. Or is it more subjective and situational? Is the particle speeding up or slowing down at time \(t=1\)? Will there be any equations where we can find the other variables (time, distance, etc) where the acceleration is not constant? Find the maximum and minimum values by using derivatives for y = x^3 (3x - 1)^2. ve(t) = 30i + (3 9.8t)j. ; 3.4.2 Calculate the average rate of change and explain how it differs from the instantaneous rate of change. ), v, start subscript, x, end subscript, squared, equals, v, start subscript, 0, x, end subscript, squared, plus, 2, a, start subscript, x, end subscript, delta, x, start text, left parenthesis, S, t, a, r, t, space, w, i, t, h, space, t, h, e, space, f, o, u, r, t, h, space, k, i, n, e, m, a, t, i, c, space, f, o, r, m, u, l, a, point, right parenthesis, end text, (Algebraicallysolveforthefinalvelocity. We can use a current population, together with a growth rate, to estimate the size of a population in the future. Whichever velocity is larger is the absolute maximum. Find the gradient of the function below and the maximum value of the directional derivative at the given point z = e ^{-x} \cos y, (0, \frac{\pi}{3}) . f(x, y, z) = sqrt(x^2 + y^2 + z^2), (9, 2, -3). In other words, the particle is being accelerated in the direction opposite the direction in which it is traveling, causing \(|v(t)|\) to decrease. Let f(x, y) = x^{2e}y. In order to find where it has an absolute maximum, we plug the endpoints of the interval into the original equation for velocity, and the larger value will be the answer. Factoring the left-hand side of the equation produces \(3(t2)(t4)=0\). Find D_uf(0,pi/3). Find the maximum value of the directional derivative at the given point. A heavier object has more inertia, which is a resistance to a change in motion. \[MC(x)=C(x)=\lim_{h0}\frac{C(x+h)C(x)}{h} \nonumber \]. We have described velocity as the rate of change of position. b. Shouldn't there be a fifth kinematic formula that is missing the initial velocity? f(x, y) = 8sin(xy), (0, 3). I looked ahead and I noticed that acceleration being constant is a lot of the content ahead. a) Find the maximum rate of change of f (x, y) = ln (x^2 + y^2) at the point (-3, 2) and the direction in which it occurs. Near the surface of the Earth, yes. A high point is called a maximum (plural maxima). Velocity is the first derivative of position with respect to time. Press the "Y=" button and enter the velocity equation. Next, use \(R(100)\) to approximate \(R(101)R(100)\), the revenue obtained from the sale of the \(101^{\text{st}}\) dinner. f(x, y) = 8y*sqrt(x); (16, 5). {/eq} are constants. Take the derivative of the slope (the second derivative of the original function): This means the slope is continually getting smaller (10): traveling from left to right the slope starts out positive (the function rises), goes through zero (the flat point), and then the slope becomes negative (the function falls): A slope that gets smaller (and goes through 0) means a maximum. At t = 5, the velocity is 46. 6. In fact it is not differentiable there (as shown on the differentiable page). For small enough values of \(h\), \(f(a)\frac{f(a+h)f(a)}{h}\). Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Learn the derivative rules, and practice taking derivatives by following examples. Thus, we can state . Accessibility StatementFor more information contact us atinfo@libretexts.org. Now, Find the maximum rate of change of the function f (x, y, z) = zx - y / z at the point (1, 4, 1). f(x, y) = (9y^2)/x, (3, 5). (a) Find the directional derivative of f at (1,1) in the direction of i + 2j. If you're seeing this message, it means we're having trouble loading external resources on our website. Find the directional derivatives of f(x, y, z) = 2xy + z^2 at the point (-4, -1, 3) in the direction of the maximum rate of change of f. Find the direction of the maximum rate of change of f(x,y)= \sin(xy) at the point (1,0). We can start with the definition of acceleration, A cool way to visually derive this kinematic formula is by considering the velocity graph for an object with constant accelerationin other words, a constant slopeand starts with initial velocity. Given \(f(10)=5\) and \(f(10)=6\), estimate \(f(10.1)\). Solving this equation leads to the two possible values Learning Objectives. z = \frac{y}{x^2 + y^2}, (1, 1). How do I find absolute minimum & maximum points with differential calculus? Find the maximum rate of change of f at the given point and the direction in which it occurs. Xilinx ISE IP Core 7.1 - FFT (settings) give incorrect results, whats missing, How can you spot MWBC's (multi-wire branch circuits) in an electrical panel. Is declarative programming just imperative programming 'under the hood'? *AP & Advanced Placement Program are registered trademarks of the College Board, which was not involved in the production of, and does not endorse this site. {/eq}, where {eq}a Begin by finding \(h\). Since \(x\) represents objects, a reasonable and small value for \(h\) is 1. It is an Inflection Point ("saddle point") the slope does become zero, but it is neither a maximum nor minimum. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. (a) Find the directional derivative of the function at the given point in the direction of the vector v. h(r, s, t) = ln (3r + 6s + 9t), (1, 1, 1), v = 4i + 12j - 6k (b) Find the maximum rate of chang. Find the maximum rate of change of f at the given point and the direction in which it occurs. Why do people say a dog is 'harmless' but not 'harmful'? Take the derivative of the velocity equation with respect to time. Where the slope is zero. Direct link to Leo Michaels's post In example 4 when pluggin, Posted 3 months ago. Previous Differentials Next Definite Integrals it is less than 0, so 3/5 is a local maximum, it is greater than 0, so +1/3 is a local minimum, 8915, 8916, 8917, 8921, 8922, 8924, 8918, 8919, 8920, 8923, equal to 0, then the test fails (there may be other ways of finding out though). Find the maximum rate of change of f at the given point and the direction in which it occurs. Since our motorcyclist will still be going in the direction of motion it started with and we assumed that direction was positive, we'll choose the positive answer, Posted 7 years ago. Part (a): The velocity of the particle is The third kinematic formula can be derived by plugging in the first kinematic formula, If we start with second kinematic formula, We can expand the right hand side and get, And finally multiplying both sides by the time. Isn't the final velocity zero since it hits the ground? Find the directional derivative using f(x, y, z) = xy + z^2 . The velocity is the derivative of the position function: b. f(x,y,z)=x^2+9xz+6yz^2;\ (1,2,-1). Press "2nd," "Calc," "Max." Where is a function at a high or low point? (answer: 228 228) (b) Find the velocity of the ball when it hits the ground. d. Before we can sketch the graph of the particle, we need to know its position at the time it starts moving \((t=0)\) and at the times that it changes direction \((t=2,4)\). \end{align} ; 3.4.3 Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. The derivative of f(x) = \frac{x^4}{3} - \frac{x^5}{5} attains its maximum value at x=? f(x)=\frac{x^2}{x^2+4} at (0,0) . In addition to analyzing velocity, speed, acceleration, and position, we can use derivatives to analyze various types of populations, including those as diverse as bacteria colonies and cities. If \(R(x)\) is the revenue obtained from selling \(x\) items, then the marginal revenue \(MR(x)\) is \(MR(x)=R(x)\). (b) Find the directional derivative of f(x, y, z). Calculate the maximum rate of change of f at the given point and the direction in which it occurs. However one of the most common forms of motion, free fall, just happens to be constant acceleration. Thus Figure 2 The graphs show the yo-yo's height, velocity, and acceleration functions from 0 to 4 seconds. Calculate the average rate of change and explain how it differs from the instantaneous rate of change. The position of a particle moving along a coordinate axis is given by \(s(t)=t^39t^2+24t+4,\; t0.\). Then take an online Calculus course at StraighterLine for college credit: http://ww. Minimum velocity. Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item. You're close, but not quite there. Function: h(x,y)=7ycos(x-y) Point: (0,\frac{\pi}{3}) The gradient listed below is correct. ve(t) = v1i + (v2 9.8t)j. f(x, y, z) = 9x + 8y/z, (6, 1, -1). Let f(x,y,z) = x^3-y^3+z^3 . Find the gradient of the function f (x, y) = 5 x - 2 x^2 + y^2 + 3 y and the maximum value of the directional derivative at P(-2, 4). Since \(v(t)=s(t)=32t\), we obtain \(v(t)=64\) ft/s. Createyouraccount. Since \(3t^218t+24<0\) on \((2,4)\), the particle is moving from right to left on this interval. Setting t = 0 and using the initial velocity of the enemy missile gives. Given $s(t) = 32 + 112 t - 16 t^{2}$ then $v(t)$, being the derivative of $s(t)$, is $v(t) = 112 - 32 t$. Its position at time \(t\) is given by \(s(t)=t^34t+2\). ae(t) = 9.8j. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. \(v_{ave}=\frac{s(2)s(0)}{20}=\frac{064}{2}=32\) ft/s. Find the maximum rate of change of f at the given point and the direction in which it occurs. The speed of the object at time \(t\) is given by \(|v(t)|\). Determine the tomato's average velocity over the interval [a, a + h]. Find the maximum value of the directional derivative of f at the point (-pi, 0). Average velocity of a ball thrown up in the air. copyright 2003-2023 Homework.Study.com. (a) Compute the directional derivative of f at P in the direction of v = \left \langle 1,-1 \right \rangle (b) Find the maximal rate of, Consider the function f(x,y,z) = 3x^2y^2+2yz . All other trademarks and copyrights are the property of their respective owners. In this case, the revenue in dollars obtained by selling \(x\) barbeque dinners is given by. This leads to singularities in the higher derivatives. Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. calculus vectors Now estimate \(P(0)\), the current growth rate, using. Find the directional derivative of at (-1,2, What is the maximal directional derivative at (1,1) of the function f(x,y) = xy(1 - x - y)? B) Find the maximum rate of. Arrow just to the right of the maximum, and again press "Enter." rev2023.8.22.43591. How do I compute the acceleration at a given time? What determines the edge/boundary of a star system? a) 4sqrt(2) b) sqrt(2) c) none of the others d)2sqrt(2) e) -4sqrt(2), a) The maximum rate of change of f(x, y, z)=tan x+ \sqrt {3} {z} at (\pi /4,3,1). Get access to this video and our entire Q&A library. A maximum is a high point and a minimum is a low point: In a smoothly changing function a maximum or minimum is always where the function flattens out (except for a saddle point). ): $v(t)=112-16t=0 \implies t=7$, then I substituted this $t$ into $s(t)$ to get $32$, which is wrong. h(0,\frac{\pi}{3})=(\frac{. { "3.4E:_Exercises_for_Section_3.4" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1]()" }, { "3.00:_Prelude_to_Derivatives" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1]()", "3.01:_Defining_the_Derivative" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1]()", "3.02:_The_Derivative_as_a_Function" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1]()", "3.03:_Differentiation_Rules" : "property get [Map 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source@https://openstax.org/details/books/calculus-volume-1. We can visually understand these three simply, you probably already have an accurate intuition of how each should look like. Let the velocity of a body be defined as a function {eq}v\left( t \right) = a{t^2} + bt Find the maximum rate of change of f at the given point and the direction in which it occurs. From the given conditions, you find that a ( t) = 4 t m/sec 2, v 0 = 0 m/sec because it begins at rest, and s 0 = -35 m because the missile is below ground level; hence, After 6 seconds, you find that hence, the missile will be 109 m above the ground after 6 seconds. Your real answer for time may likely involve . Divide the decimal time by . (b) Find the maximum rate of change of the f(x, y) at the point (3, (1) Find the gradient of the function and the maximum value of the directional derivative at the point. Similarly, we can find the location of any relative maxima or minima of a function by setting the derivative equal to zero. Find the maximum rate of change of f at the given point and the direction in which it occurs. Find (a) the gradient of the function and (b) the maximum values of the directional derivative of z = ln(x^2-y) at the point (2, 3). Why is the time interval now written as t? b) The directional derivative of f(x, y, z)=e^{xy} + z at the point (1,1,0) in the direction of 2 \vec i-3 \vec j+4 \. The best answers are voted up and rise to the top, Not the answer you're looking for? Use \(P(100)\) to approximate \(P(101)P(100)\). Direct link to Mark Zwald's post Near the surface of the E, Posted 6 years ago. g(x, y) = \ln \sqrt[3]{x^2 + y^2} at (1, 2). Find the directional derivative of f(x,y) = 2 \sqrt x - y^2 at the point (1,5) in the direction toward the point (4,1) Part 2. If \(C(x)\) is the cost of producing \(x\) items, then the. The general word for maximum or minimum is extremum (plural extrema). z = x^3 y^2 at (3, 2), Find the partial derivative with respect to x and the global maximum of f(x,y) = x^2y^x, . (2) Find the unit vector into which the rate change occurs. When the slope is equal to zero, the line is horizontal. (b) Find the directional derivative of f at the point P(1,2, 1) in direction from P to A(3,0,0). delta, x, start text, D, i, s, p, l, a, c, e, m, e, n, t, end text, t, start text, T, i, m, e, space, i, n, t, e, r, v, a, l, end text, space, v, start subscript, 0, end subscript, space, space, start text, I, n, i, t, i, a, l, space, v, e, l, o, c, i, t, y, end text, space, v, space, space, space, start text, F, i, n, a, l, space, v, e, l, o, c, i, t, y, end text, space, a, space, space, start text, space, C, o, n, s, t, a, n, t, space, a, c, c, e, l, e, r, a, t, i, o, n, end text, space, delta, x, comma, t, comma, v, start subscript, 0, end subscript, comma, v, comma, a, 1, point, v, equals, v, start subscript, 0, end subscript, plus, a, t, 2, point, delta, x, equals, left parenthesis, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, right parenthesis, t, 3, point, delta, x, equals, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, a, t, squared, 4, point, v, squared, equals, v, start subscript, 0, end subscript, squared, plus, 2, a, delta, x, g, equals, 9, point, 81, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, (Magnitudeofaccelerationduetogravity), g, equals, 9, point, 81, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, start text, left parenthesis, M, a, g, n, i, t, u, d, e, space, o, f, space, a, c, c, e, l, e, r, a, t, i, o, n, space, d, u, e, space, t, o, space, g, r, a, v, i, t, y, right parenthesis, end text, a, start subscript, y, end subscript, equals, minus, 9, point, 81, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, v, start subscript, 0, end subscript, equals, 5, start text, space, m, slash, s, end text, t, equals, 3, start text, space, s, end text, delta, x, equals, 8, start text, space, m, end text, delta, x, equals, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, a, t, squared, delta, x, comma, v, start subscript, 0, end subscript, comma, t, 1, point, v, equals, v, start subscript, 0, end subscript, plus, a, t, start text, left parenthesis, T, h, i, s, space, f, o, r, m, u, l, a, space, i, s, space, m, i, s, s, i, n, g, space, delta, x, point, right parenthesis, end text, 2, point, delta, x, equals, left parenthesis, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, right parenthesis, t, start text, left parenthesis, T, h, i, s, space, f, o, r, m, u, l, a, space, i, s, space, m, i, s, s, i, n, g, space, a, point, right parenthesis, end text, 3, point, delta, x, equals, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, a, t, squared, start text, left parenthesis, T, h, i, s, space, f, o, r, m, u, l, a, space, i, s, space, m, i, s, s, i, n, g, space, v, point, right parenthesis, end text, 4, point, v, squared, equals, v, start subscript, 0, end subscript, squared, plus, 2, a, delta, x, start text, left parenthesis, T, h, i, s, space, f, o, r, m, u, l, a, space, i, s, space, m, i, s, s, i, n, g, space, t, point, right parenthesis, end text, v, equals, v, start subscript, 0, end subscript, plus, a, t, a, equals, start fraction, delta, v, divided by, delta, t, end fraction, v, minus, v, start subscript, 0, end subscript, a, equals, start fraction, v, minus, v, start subscript, 0, end subscript, divided by, delta, t, end fraction, v, equals, v, start subscript, 0, end subscript, plus, a, delta, t, delta, x, equals, left parenthesis, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, right parenthesis, t, delta, x, equals, start text, space, t, o, t, a, l, space, a, r, e, a, end text, start fraction, 1, divided by, 2, end fraction, t, left parenthesis, v, minus, v, start subscript, 0, end subscript, right parenthesis, delta, x, equals, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, t, left parenthesis, v, minus, v, start subscript, 0, end subscript, right parenthesis, start fraction, 1, divided by, 2, end fraction, t, delta, x, equals, v, start subscript, 0, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, v, t, minus, start fraction, 1, divided by, 2, end fraction, v, start subscript, 0, end subscript, t, delta, x, equals, start fraction, 1, divided by, 2, end fraction, v, t, plus, start fraction, 1, divided by, 2, end fraction, v, start subscript, 0, end subscript, t, start fraction, delta, x, divided by, t, end fraction, equals, left parenthesis, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, right parenthesis, start fraction, delta, x, divided by, t, end fraction, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, A, start subscript, r, e, c, t, a, n, g, l, e, end subscript, equals, h, w, start fraction, 1, divided by, 2, end fraction, a, t, squared, A, start subscript, t, r, i, a, n, g, l, e, end subscript, equals, start fraction, 1, divided by, 2, end fraction, b, h, start fraction, delta, x, divided by, t, end fraction, equals, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, start fraction, delta, x, divided by, t, end fraction, equals, start fraction, left parenthesis, v, start subscript, 0, end subscript, plus, a, t, right parenthesis, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, start fraction, delta, x, divided by, t, end fraction, equals, start fraction, v, start subscript, 0, end subscript, divided by, 2, end fraction, plus, start fraction, a, t, divided by, 2, end fraction, plus, start fraction, v, start subscript, 0, end subscript, divided by, 2, end fraction, start fraction, v, start subscript, 0, end subscript, divided by, 2, end fraction, start fraction, delta, x, divided by, t, end fraction, equals, v, start subscript, 0, end subscript, plus, start fraction, a, t, divided by, 2, end fraction, v, squared, equals, v, start subscript, 0, end subscript, squared, plus, 2, a, delta, x, t, equals, start fraction, v, minus, v, start subscript, 0, end subscript, divided by, a, end fraction, delta, x, equals, left parenthesis, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, right parenthesis, left parenthesis, start fraction, v, minus, v, start subscript, 0, end subscript, divided by, a, end fraction, right parenthesis, delta, x, equals, left parenthesis, start fraction, v, squared, minus, v, start subscript, 0, end subscript, squared, divided by, 2, a, end fraction, right parenthesis, v, start subscript, 0, end subscript, equals, 0, delta, x, comma, v, start subscript, o, end subscript, comma, v, comma, a, a, start subscript, g, end subscript, equals, minus, 9, point, 81, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, t, equals, 2, point, 35, start text, space, s, end text, (Usethefirstkinematicformulasinceitsmissing, v, equals, v, start subscript, 0, end subscript, plus, a, t, start text, left parenthesis, U, s, e, space, t, h, e, space, f, i, r, s, t, space, k, i, n, e, m, a, t, i, c, space, f, o, r, m, u, l, a, space, s, i, n, c, e, space, i, t, apostrophe, s, space, m, i, s, s, i, n, g, space, delta, y, point, right parenthesis, end text, v, equals, 0, start text, space, m, slash, s, end text, plus, left parenthesis, minus, 9, point, 81, start fraction, start text, m, end text, divided by, start text, s, end text, squared, end fraction, right parenthesis, left parenthesis, 2, point, 35, start text, space, s, end text, right parenthesis, start text, left parenthesis, P, l, u, g, space, i, n, space, k, n, o, w, n, space, v, a, l, u, e, s, point, right parenthesis, end text, v, equals, minus, 23, point, 1, start text, space, m, slash, s, end text, start text, left parenthesis, C, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, !, right parenthesis, end text, v, start subscript, 0, end subscript, equals, 6, point, 20, start text, space, m, slash, s, end text, v, equals, 23, point, 1, start text, space, m, slash, s, end text, t, equals, 3, point, 30, start text, space, s, end text, (Usethesecondkinematicformulasinceitsmissing, delta, x, equals, left parenthesis, start fraction, v, plus, v, start subscript, 0, end subscript, divided by, 2, end fraction, right parenthesis, t, start text, left parenthesis, U, s, e, space, t, h, e, space, s, e, c, o, n, d, space, k, i, n, e, m, a, t, i, c, space, f, o, r, m, u, l, a, space, s, i, n, c, e, space, i, t, apostrophe, s, space, m, i, s, s, i, n, g, space, a, point, right parenthesis, end text, delta, x, equals, left parenthesis, start fraction, 23, point, 1, start text, space, m, slash, s, end text, plus, 6, point, 20, start text, space, m, slash, s, end text, divided by, 2, end fraction, right parenthesis, left parenthesis, 3, point, 30, start text, space, s, end text, right parenthesis, start text, left parenthesis, P, l, u, g, space, i, n, space, k, n, o, w, n, space, v, a, l, u, e, s, point, right parenthesis, end text, delta, x, equals, 48, point, 3, start text, space, m, end text, start text, left parenthesis, C, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, !, right parenthesis, end text, v, start subscript, 0, end subscript, equals, 18, point, 3, start text, space, m, slash, s, end text, delta, y, equals, 12, point, 2, start text, space, m, end text, a, equals, minus, 9, point, 81, start fraction, start text, space, m, end text, divided by, start text, space, s, end text, squared, end fraction, delta, y, equals, v, start subscript, 0, y, end subscript, t, plus, start fraction, 1, divided by, 2, end fraction, a, start subscript, y, end subscript, t, squared, (Startwiththethirdkinematicformula.
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