The derivative of a function of real variable represents how a function changes in response to the change in another variable. One of its application is used in solving problems related to dynamics of rigid bodies and in determination of forces and strength of . Solving the initial value problem \[ \frac{dy}{dx} = f(x), \mbox{ with the initial condition } y(x_{0}) = y_{0} \] requires you to: first find the set of antiderivatives of \( f \) and then. A method for approximating the roots of \( f(x) = 0 \). 3. Let \( c \) be a critical point of a function \( f. \)What does The Second Derivative Test tells us if \( f''(c) >0 \)? There are lots of different articles about related rates, including Rates of Change, Motion Along a Line, Population Change, and Changes in Cost and Revenue. StudySmarter is commited to creating, free, high quality explainations, opening education to all. The limit of the function \( f(x) \) is \( - \infty \) as \( x \to \infty \) if \( f(x) < 0 \) and \( \left| f(x) \right| \) becomes larger and larger as \( x \) also becomes larger and larger. A function can have more than one local minimum. Learn derivatives of cos x, derivatives of sin x, derivatives of xsinx and derivative of 2x here. Now by substituting the value of dx/dt and dy/dt in the above equation we get, \(\Rightarrow \frac{{dA}}{{dt}} = \left( { \;5} \right) \cdot y + x \cdot 6\). Application of derivatives Class 12 notes is about finding the derivatives of the functions. We can state that at x=c if f(x)f(c) for every value of x in the domain we are operating on, then f(x) has an absolute minimum; this is also known as the global minimum value. Since you want to find the maximum possible area given the constraint of \( 1000ft \) of fencing to go around the perimeter of the farmland, you need an equation for the perimeter of the rectangular space. Many engineering principles can be described based on such a relation. Now lets find the roots of the equation f'(x) = 0, Now lets find out f(x) i.e \(\frac{d^2(f(x))}{dx^2}\), Now evaluate the value of f(x) at x = 12, As we know that according to the second derivative test if f(c) < 0 then x = c is a point of maxima, Hence, the required numbers are 12 and 12. Derivatives play a very important role in the world of Mathematics. Aerospace Engineers could study the forces that act on a rocket. Exponential and Logarithmic functions; 7. We use the derivative to determine the maximum and minimum values of particular functions (e.g. Before jumping right into maximizing the area, you need to determine what your domain is. While quite a major portion of the techniques is only useful for academic purposes, there are some which are important in the solution of real problems arising from science and engineering. Find an equation that relates your variables. The derivative of the given curve is: \[ f'(x) = 2x \], Plug the \( x \)-coordinate of the given point into the derivative to find the slope.\[ \begin{align}f'(x) &= 2x \\f'(2) &= 2(2) \\ &= 4 \\ &= m.\end{align} \], Use the point-slope form of a line to write the equation.\[ \begin{align}y-y_1 &= m(x-x_1) \\y-4 &= 4(x-2) \\y &= 4(x-2)+4 \\ &= 4x - 4.\end{align} \]. At an instant t, let its radius be r and surface area be S. As we know the surface area of a sphere is given by: 4r2where r is the radius of the sphere. One of the most important theorems in calculus, and an application of derivatives, is the Mean Value Theorem (sometimes abbreviated as MVT). What is the maximum area? The absolute minimum of a function is the least output in its range. State Corollary 2 of the Mean Value Theorem. Ltd.: All rights reserved. Earn points, unlock badges and level up while studying. Applications of the Derivative 1. The slope of a line tangent to a function at a critical point is equal to zero. project. At its vertex. Write a formula for the quantity you need to maximize or minimize in terms of your variables. Plugging this value into your revenue equation, you get the \( R(p) \)-value of this critical point:\[ \begin{align}R(p) &= -6p^{2} + 600p \\R(50) &= -6(50)^{2} + 600(50) \\R(50) &= 15000.\end{align} \]. BASIC CALCULUS | 4TH GRGADING PRELIM APPLICATION OF DERIVATIVES IN REAL LIFE The derivative is the exact rate at which one quantity changes with respect to another. The Quotient Rule; 5. Solution:Here we have to find the rate of change of the area of a circle with respect to its radius r when r = 6 cm. Quality and Characteristics of Sewage: Physical, Chemical, Biological, Design of Sewer: Types, Components, Design And Construction, More, Approximation or Finding Approximate Value, Equation of a Tangent and Normal To a Curve, Determining Increasing and Decreasing Functions. These results suggest that cell-seeding onto chitosan-based scaffolds would provide tissue engineered implant being biocompatible and viable. in an electrical circuit. Solution: Given f ( x) = x 2 x + 6. The normal line to a curve is perpendicular to the tangent line. A function is said to be concave down, or concave, in an interval where: A function is said to be concave up, or convex, in an interval where: An x-value for which the concavity of a graph changes. Data science has numerous applications for organizations, but here are some for mechanical engineering: 1. If \( f'(x) < 0 \) for all \( x \) in \( (a, b) \), then \( f \) is a decreasing function over \( [a, b] \). Some of them are based on Minimal Cut (Path) Sets, which represent minimal sets of basic events, whose simultaneous occurrence leads to a failure (repair) of the . In recent years, great efforts have been devoted to the search for new cost-effective adsorbents derived from biomass. So, by differentiating A with respect to twe get: \(\frac{{dA}}{{dt}} = \frac{{dA}}{{dr}} \cdot \frac{{dr}}{{dt}}\) (Chain Rule), \(\Rightarrow \frac{{dA}}{{dr}} = \frac{{d\left( { \cdot {r^2}} \right)}}{{dr}} = 2 r\), \(\Rightarrow \frac{{dA}}{{dt}} = 2 r \cdot \frac{{dr}}{{dt}}\), By substituting r = 6 cm and dr/dt = 8 cm/sec in the above equation we get, \(\Rightarrow \frac{{dA}}{{dt}} = 2 \times 6 \times 8 = 96 \;c{m^2}/sec\). The only critical point is \( x = 250 \). You can use LHpitals rule to evaluate the limit of a quotient when it is in either of the indeterminate forms \( \frac{0}{0}, \ \frac{\infty}{\infty} \). This formula will most likely involve more than one variable. If y = f(x), then dy/dx denotes the rate of change of y with respect to xits value at x = a is denoted by: Decreasing rate is represented by negative sign whereas increasing rate is represented bypositive sign. Sign In. Chitosan and its derivatives are polymers made most often from the shells of crustaceans . Computer algebra systems that compute integrals and derivatives directly, either symbolically or numerically, are the most blatant examples here, but in addition, any software that simulates a physical system that is based on continuous differential equations (e.g., computational fluid dynamics) necessarily involves computing derivatives and . The tangent line to a curve is one that touches the curve at only one point and whose slope is the derivative of the curve at that point. Example 8: A stone is dropped into a quite pond and the waves moves in circles. ENGR 1990 Engineering Mathematics Application of Derivatives in Electrical Engineering The diagram shows a typical element (resistor, capacitor, inductor, etc.) A point where the derivative (or the slope) of a function is equal to zero. The key concepts and equations of linear approximations and differentials are: A differentiable function, \( y = f(x) \), can be approximated at a point, \( a \), by the linear approximation function: Given a function, \( y = f(x) \), if, instead of replacing \( x \) with \( a \), you replace \( x \) with \( a + dx \), then the differential: is an approximation for the change in \( y \). Taking partial d Example 6: The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate 4 cm/minute. The normal is a line that is perpendicular to the tangent obtained. So, by differentiating S with respect to t we get, \(\Rightarrow \frac{{dS}}{{dt}} = \frac{{dS}}{{dr}} \cdot \frac{{dr}}{{dt}}\), \(\Rightarrow \frac{{dS}}{{dr}} = \frac{{d\left( {4 {r^2}} \right)}}{{dr}} = 8 r\), By substituting the value of dS/dr in dS/dt we get, \(\Rightarrow \frac{{dS}}{{dt}} = 8 r \cdot \frac{{dr}}{{dt}}\), By substituting r = 5 cm, = 3.14 and dr/dt = 0.02 cm/sec in the above equation we get, \(\Rightarrow {\left[ {\frac{{dS}}{{dt}}} \right]_{r = 5}} = \left( {8 \times 3.14 \times 5 \times 0.02} \right) = 2.512\;c{m^2}/sec\). Now if we consider a case where the rate of change of a function is defined at specific values i.e. Hence, therate of change of the area of a circle with respect to its radius r when r = 6 cm is 12 cm. Similarly, f(x) is said to be a decreasing function: As we know that,\(\frac{{d\left( {{{\tan }^{ 1}}x} \right)}}{{dx}} = \frac{1}{{1 + {x^2}}}\;\)and according to chain rule\(\frac{{dy}}{{dx}} = \frac{{dy}}{{dv}} \cdot \frac{{dv}}{{dx}}\), \( f\left( x \right) = \frac{1}{{1 + {{\left( {\cos x + \sin x} \right)}^2}}} \cdot \frac{{d\left( {\cos x + \sin x} \right)}}{{dx}}\), \( f\left( x \right) = \frac{{\cos x \sin x}}{{2 + \sin 2x}}\), Now when 0 < x sin x and sin 2x > 0, As we know that for a strictly increasing function f'(x) > 0 for all x (a, b). Principal steps in reliability engineering include estimation of system reliability and identification and quantification of situations which cause a system failure. Examples on how to apply and use inverse functions in real life situations and solve problems in mathematics. Then \(\frac{dy}{dx}\) denotes the rate of change of y w.r.t x and its value at x = a is denoted by: \(\left[\frac{dy}{dx}\right]_{_{x=a}}\). Now by differentiating V with respect to t, we get, \( \frac{{dV}}{{dt}} = \frac{{dV}}{{dx}} \cdot \frac{{dx}}{{dt}}\)(BY chain Rule), \( \frac{{dV}}{{dx}} = \frac{{d\left( {{x^3}} \right)}}{{dx}} = 3{x^2}\). What is the absolute minimum of a function? For such a cube of unit volume, what will be the value of rate of change of volume? Let y = f(v) be a differentiable function of v and v = g(x) be a differentiable function of x then. In this article, you will discover some of the many applications of derivatives and how they are used in calculus, engineering, and economics. A differential equation is the relation between a function and its derivatives. Related Rates 3. Create the most beautiful study materials using our templates. In many applications of math, you need to find the zeros of functions. Create and find flashcards in record time. Both of these variables are changing with respect to time. This Class 12 Maths chapter 6 notes contains the following topics: finding the derivatives of the equations, rate of change of quantities, Increasing and decreasing functions, Tangents and normal, Approximations, Maxima and minima, and many more. Then the rate of change of y w.r.t x is given by the formula: \(\frac{y}{x}=\frac{y_2-y_1}{x_2-x_1}\). Biomechanical. No. As we know that,\(\frac{d}{{dx}}\left[ {f\left( x \right) \cdot g\left( x \right)} \right] = f\left( x \right) \cdot \;\frac{{d\left\{ {g\left( x \right)} \right\}}}{{dx}}\; + \;\;g\left( x \right) \cdot \;\frac{{d\left\{ {f\left( x \right)} \right\}}}{{dx}}\). Determine for what range of values of the other variables (if this can be determined at this time) you need to maximize or minimize your quantity. To name a few; All of these engineering fields use calculus. Partial differential equations such as that shown in Equation (2.5) are the equations that involve partial derivatives described in Section 2.2.5. The applications of this concept in the field of the engineering are spread all over engineering subjects and sub-fields ( Taylor series ). Use these equations to write the quantity to be maximized or minimized as a function of one variable. Your camera is \( 4000ft \) from the launch pad of a rocket. Then dy/dx can be written as: \(\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\left(\frac{d y}{d t} \cdot \frac{d t}{d x}\right)\)with the help of chain rule. Derivative is the slope at a point on a line around the curve. Applications of Derivatives in Optimization Algorithms We had already seen that an optimization algorithm, such as gradient descent, seeks to reach the global minimum of an error (or cost) function by applying the use of derivatives. Set individual study goals and earn points reaching them. Will you pass the quiz? Applications of Derivatives in Various fields/Sciences: Such as in: -Physics -Biology -Economics -Chemistry -Mathematics -Others(Psychology, sociology & geology) 15. cost, strength, amount of material used in a building, profit, loss, etc.). Once you understand derivatives and the shape of a graph, you can build on that knowledge to graph a function that is defined on an unbounded domain. If \( f \) is differentiable over \( I \), except possibly at \( c \), then \( f(c) \) satisfies one of the following: If \( f' \) changes sign from positive when \( x < c \) to negative when \( x > c \), then \( f(c) \) is a local max of \( f \). Assign symbols to all the variables in the problem and sketch the problem if it makes sense. Newton's method approximates the roots of \( f(x) = 0 \) by starting with an initial approximation of \( x_{0} \). They all use applications of derivatives in their own way, to solve their problems. You want to record a rocket launch, so you place your camera on your trusty tripod and get it all set up to record this event. If \( f(c) \leq f(x) \) for all \( x \) in the domain of \( f \), then you say that \( f \) has an absolute minimum at \( c \). Civil Engineers could study the forces that act on a bridge. One of many examples where you would be interested in an antiderivative of a function is the study of motion. What are practical applications of derivatives? Industrial Engineers could study the forces that act on a plant. If the company charges \( $20 \) or less per day, they will rent all of their cars. A critical point of the function \( g(x)= 2x^3+x^2-1\) is \( x=0. a specific value of x,. The Chain Rule; 4 Transcendental Functions. Engineering Application of Derivative in Different Fields Michael O. Amorin IV-SOCRATES Applications and Use of the Inverse Functions. These extreme values occur at the endpoints and any critical points. Derivatives help business analysts to prepare graphs of profit and loss. The second derivative of a function is \( g''(x)= -2x.\) Is it concave or convex at \( x=2 \)? Now, if x = f(t) and y = g(t), suppose we want to find the rate of change of y concerning x. b): x Fig. 8.1 INTRODUCTION This chapter will discuss what a derivative is and why it is important in engineering. Variables whose variations do not depend on the other parameters are 'Independent variables'. Biomechanical Applications Drug Release Process Numerical Methods Back to top Authors and Affiliations College of Mechanics and Materials, Hohai University, Nanjing, China Wen Chen, HongGuang Sun School of Mathematical Sciences, University of Jinan, Jinan, China Xicheng Li Back to top About the authors of the users don't pass the Application of Derivatives quiz! It provided an answer to Zeno's paradoxes and gave the first . The two main applications that we'll be looking at in this chapter are using derivatives to determine information about graphs of functions and optimization problems. a one-dimensional space) and so it makes some sense then that when integrating a function of two variables we will integrate over a region of (two dimensional space). Similarly, we can get the equation of the normal line to the curve of a function at a location. Here we have to find therate of change of the area of a circle with respect to its radius r when r = 6 cm. Even the financial sector needs to use calculus! Letf be a function that is continuous over [a,b] and differentiable over (a,b). Additionally, you will learn how derivatives can be applied to: Derivatives are very useful tools for finding the equations of tangent lines and normal lines to a curve. Derivative further finds application in the study of seismology to detect the range of magnitudes of the earthquake. As we know that, ify = f(x), then dy/dx denotes the rate of change of y with respect to x. You use the tangent line to the curve to find the normal line to the curve. The degree of derivation represents the variation corresponding to a "speed" of the independent variable, represented by the integer power of the independent variation. The problem has four design variables: {T_s}= {x_1} thickness of shell, {T_h}= {x_2} thickness of head, R= {x_3} inner radius, and L= {x_4} length of cylindrical section of vessel Fig. There are many important applications of derivative. You must evaluate \( f'(x) \) at a test point \( x \) to the left of \( c \) and a test point \( x \) to the right of \( c \) to determine if \( f \) has a local extremum at \( c \). At any instant t, let A be the area of rectangle, x be the length of the rectangle and y be the width of the rectangle. The function \( f(x) \) becomes larger and larger as \( x \) also becomes larger and larger. How do I study application of derivatives? What are the applications of derivatives in economics? If you make substitute the known values before you take the derivative, then the substituted quantities will behave as constants and their derivatives will not appear in the new equation you find in step 4. Example 1: Find the rate of change of the area of a circle with respect to its radius r when r = 6 cm. The basic applications of double integral is finding volumes. To maximize revenue, you need to balance the price charged per rental car per day against the number of cars customers will rent at that price. If a function, \( f \), has a local max or min at point \( c \), then you say that \( f \) has a local extremum at \( c \). Every local maximum is also a global maximum. An antiderivative of a function \( f \) is a function whose derivative is \( f \). Upload unlimited documents and save them online. The purpose of this application is to minimize the total cost of design, including the cost of the material, forming, and welding. Meanwhile, futures and forwards contracts, swaps, warrants, and options are the most widely used types of derivatives. Calculus In Computer Science. Under this heading, we will use applications of derivatives and methods of differentiation to discover whether a function is increasing, decreasing or none. Using the chain rule, take the derivative of this equation with respect to the independent variable. derivatives are the functions required to find the turning point of curve What is the role of physics in electrical engineering? A function can have more than one global maximum. For more information on this topic, see our article on the Amount of Change Formula. This area of interest is important to many industriesaerospace, defense, automotive, metals, glass, paper and plastic, as well as to the thermal design of electronic and computer packages. For a function f defined on an interval I the maxima or minima ( or local maxima or local minima) in I depends on the given condition: f(x) f(c) or f (x) f(c), x I and c is a point in the interval I. application of partial . To find \( \frac{d \theta}{dt} \), you first need to find \(\sec^{2} (\theta) \). \({\left[ {\frac{{dy}}{{dx}}} \right]_{x = a}}\), \(\frac{{dy}}{{dx}} = \frac{{dy}}{{dv}} \cdot \frac{{dv}}{{dx}}\), \( \frac{{dV}}{{dt}} = \frac{{dV}}{{dx}} \cdot \frac{{dx}}{{dt}}\), \( \frac{{dV}}{{dt}} = 3{x^2} \cdot \frac{{dx}}{{dt}}\), \(\Rightarrow \frac{{dV}}{{dt}} = 3{x^2} \cdot 5 = 15{x^2}\), \(\Rightarrow {\left[ {\frac{{dV}}{{dt}}} \right]_{x = 10}} = 15 \times {10^2} = 1500\;c{m^3}/sec\), \(\frac{d}{{dx}}\left[ {f\left( x \right) \cdot g\left( x \right)} \right] = f\left( x \right) \cdot \;\frac{{d\left\{ {g\left( x \right)} \right\}}}{{dx}}\; + \;\;g\left( x \right) \cdot \;\frac{{d\left\{ {f\left( x \right)} \right\}}}{{dx}}\), \(\frac{{dA}}{{dt}} = \frac{{dA}}{{dr}} \cdot \frac{{dr}}{{dt}}\), \({\left[ {\frac{{dA}}{{dr}}} \right]_{r\; = 6}}\), \(\frac{{d\left( {{{\tan }^{ 1}}x} \right)}}{{dx}} = \frac{1}{{1 + {x^2}}}\;\), \(\frac{{dy}}{{dx}} > 0\;or\;f\left( x \right) > 0\), \(\frac{{dy}}{{dx}} < 0\;or\;f\left( x \right) < 0\), \(\frac{{dy}}{{dx}} \ge 0\;or\;f\left( x \right) \ge 0\), \(\frac{{dy}}{{dx}} \le 0\;or\;f\left( x \right) \le 0\). Chitosan derivatives for tissue engineering applications. The key terms and concepts of antiderivatives are: A function \( F(x) \) such that \( F'(x) = f(x) \) for all \( x \) in the domain of \( f \) is an antiderivative of \( f \). Engineering Application Optimization Example. When x = 8 cm and y = 6 cm then find the rate of change of the area of the rectangle. Optimization 2. Rolle's Theorem says that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a,b), andf(a)=f(b), then there is at least one valuecwheref'(c)= 0. Suggested courses (NOTE: courses are approved to satisfy Restricted Elective requirement): Aerospace Science and Engineering 138; Mechanical Engineering . Determine the dimensions \( x \) and \( y \) that will maximize the area of the farmland using \( 1000ft \) of fencing. Derivatives are met in many engineering and science problems, especially when modelling the behaviour of moving objects. Let \(x_1, x_2\) be any two points in I, where \(x_1, x_2\) are not the endpoints of the interval. Now if we say that y changes when there is some change in the value of x. \]. The normal is perpendicular to the tangent therefore the slope of normal at any point say is given by: \(-\frac{1}{\text{Slopeoftangentatpoint}\ \left(x_1,\ y_1\ \right)}=-\frac{1}{m}=-\left[\frac{dx}{dy}\right]_{_{\left(x_1,\ y_1\ \right)}}\). Therefore, the maximum revenue must be when \( p = 50 \). Free and expert-verified textbook solutions. They have a wide range of applications in engineering, architecture, economics, and several other fields. We also allow for the introduction of a damper to the system and for general external forces to act on the object. However, you don't know that a function necessarily has a maximum value on an open interval, but you do know that a function does have a max (and min) value on a closed interval. You are an agricultural engineer, and you need to fence a rectangular area of some farmland. If you think about the rocket launch again, you can say that the rate of change of the rocket's height, \( h \), is related to the rate of change of your camera's angle with the ground, \( \theta \). Using the derivative to find the tangent and normal lines to a curve. Once you learn the methods of finding extreme values (also known collectively as extrema), you can apply these methods to later applications of derivatives, like creating accurate graphs and solving optimization problems. Water pollution by heavy metal ions is currently of great concern due to their high toxicity and carcinogenicity. Derivative is the slope at a point on a line around the curve. You find the application of the second derivative by first finding the first derivative, then the second derivative of a function. If the parabola opens upwards it is a minimum. Learn. Since the area must be positive for all values of \( x \) in the open interval of \( (0, 500) \), the max must occur at a critical point. look for the particular antiderivative that also satisfies the initial condition. Identify your study strength and weaknesses. Similarly, we can get the equation of the normal line to the curve of a function at a location. Stationary point of the function \(f(x)=x^2x+6\) is 1/2. The collaboration effort involved enhancing the first year calculus courses with applied engineering and science projects. An increasing function's derivative is. Newton's Methodis a recursive approximation technique for finding the root of a differentiable function when other analytical methods fail. Wow - this is a very broad and amazingly interesting list of application examples. Let \( c \) be a critical point of a function \( f. \)What does The Second Derivative Test tells us if \( f''(c)=0 \)? The above formula is also read as the average rate of change in the function. An example that is common among several engineering disciplines is the use of derivatives to study the forces acting on an object. State Corollary 3 of the Mean Value Theorem. Applications of Derivatives in Maths The derivative is defined as the rate of change of one quantity with respect to another. Solved Examples Example 4: Find the Stationary point of the function \(f(x)=x^2x+6\), As we know that point c from the domain of the function y = f(x) is called the stationary point of the function y = f(x) if f(c)=0. How do I find the application of the second derivative? Since biomechanists have to analyze daily human activities, the available data piles up . Now we have to find the value of dA/dr at r = 6 cm i.e\({\left[ {\frac{{dA}}{{dr}}} \right]_{r\; = 6}}\), \(\Rightarrow {\left[ {\frac{{dA}}{{dr}}} \right]_{r\; = 6}} = 2 \cdot 6 = 12 \;cm\). b Calculus is one of the most important breakthroughs in modern mathematics, answering questions that had puzzled mathematicians, scientists, and philosophers for more than two thousand years. Well, this application teaches you how to use the first and second derivatives of a function to determine the shape of its graph. This means you need to find \( \frac{d \theta}{dt} \) when \( h = 1500ft \). Derivative of a function can further be applied to determine the linear approximation of a function at a given point. Linearity of the Derivative; 3. How do you find the critical points of a function? Similarly, at x=c if f(x)f(c) for every value of x on some open interval, say (r, s), then f(x) has a relative minimum; this is also known as the local minimum value. b) 20 sq cm. The greatest value is the global maximum. When it comes to functions, linear functions are one of the easier ones with which to work. Let \( f \) be differentiable on an interval \( I \). In terms of functions, the rate of change of function is defined as dy/dx = f (x) = y'. Hence, the rate of change of the area of a circle with respect to its radius r when r = 6 cm is 12 cm. Mathematical optimizationis the study of maximizing or minimizing a function subject to constraints, essentially finding the most effective and functional solution to a problem. If \( f''(x) < 0 \) for all \( x \) in \( I \), then \( f \) is concave down over \( I \). 1. Newton's method saves the day in these situations because it is a technique that is efficient at approximating the zeros of functions. This is called the instantaneous rate of change of the given function at that particular point. If you have mastered Applications of Derivatives, you can learn about Integral Calculus here. In the times of dynamically developing regenerative medicine, more and more attention is focused on the use of natural polymers. Example 9: Find the rate of change of the area of a circle with respect to its radius r when r = 6 cm. A relative maximum of a function is an output that is greater than the outputs next to it. Computationally, partial differentiation works the same way as single-variable differentiation with all other variables treated as constant. When the slope of the function changes from +ve to -ve moving via point c, then it is said to be maxima. To accomplish this, you need to know the behavior of the function as \( x \to \pm \infty \). Given: dx/dt = 5cm/minute and dy/dt = 4cm/minute. As we know that slope of the tangent at any point say \((x_1, y_1)\) to a curve is given by: \(m=\left[\frac{dy}{dx}\right]_{_{(x_1,y_1)}}\), \(m=\left[\frac{dy}{dx}\right]_{_{(1,3)}}=(4\times1^318\times1^2+26\times110)=2\).
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