The top of the ladder is dropping at a rate of 4 in/s. THE CALCULUS OF DIFFERENTIAL FORMS 305 Chapter 39. Solve the differential equation: x'' + 5x' + 4x = 0. Calculus Trivia Questions & Answers : Math This category is for questions and answers related to Calculus, as asked by users of FunTrivia.com. add. Submit order. Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis? Differential Calculus Quizzes Check your mastery of this concept by taking a short quiz. In this problem, we will solve the initial value inhomogeneous differential equation in two steps. Determine whether or not F is a conservative vector field. Browse through all study tools. Show that y = Cex3, where C is a real number, is a solution of the following differential equation y' - 3x2y = 0. y = Ct^{-3}; ty'(t) + 3y = 0. ... Related Calculus Q&A. x'' + 36 x = 9 cos (7 t), x (0) = 5, x' (0) = 1. Here is a set of practice problems to accompany the Differentiation Formulas section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. A 5 m long ladder is leaning against a wall. Date Rating. (Give your answers correct to 2 decimal places.) A comprehensive database of differential calculus quizzes online, test your knowledge with differential calculus quiz questions. The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x) = 1000 + 32x - 2x^2. Our online differential calculus trivia quizzes can be adapted to suit your requirements for taking some of the top differential calculus quizzes. Find \Delta y. It turns out to be rather di cult to give a precise description of what a number is, and in this course we won’t try to get anywhere near the bottom of this issue. For a differentiable function f(x, y, z) compute curl (f grad f). Browse through all study tools. Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. A water tank is being drained and has the shape of a rectangular box 7m long, 6m wide and 5m high. find a potential function. Find the marginal profit at $12,000. Determine whether the vector field is conservative. A Guide to Differential Calculus Teaching Approach Calculus forms an integral part of the Mathematics Grade 12 syllabus and its applications in everyday life is widespread and important in every aspect, from being able to determine the maximum expansion and contraction of bridges to determining the maximum volume or maximum area given a function. Find the radius and height giving a minimum surface area for a tank having the shape of a right circular cylinder and a volume of 2 m^3. 2t^2 y'' + 3ty' - y = 0, Find the general solution of the given equation. y''-y'-2y=x. Find potential functions for the vector field F by inspection. How fast is the water level decreasing? Check your mastery of this concept by taking a short quiz. Answer... A company needs to make a cylindrical can that can hold precisely 1.5 liters of liquid. Solve using variation of parameters: y" - 2y' - 8y = 2e^{-3x}, Is the following differential equation separable or not? If the trough is being filled with water at a rate of 13 ft^3/min,... Find the differential of the function. Find exactly the differential of f(x,y) = \sqrt{x^2 + y^2} at the point (1,2). Find y'''(0), where y = 6 x^4 + 8 x^3 + 7. Earn Transferable Credit & Get your Degree, Concavity and Inflection Points with Differentiation, Calculating Derivatives of Trigonometric Functions, Finding Derivatives of Implicit Functions, How to Use the Chain Rule to Differentiate Complex Functions, Interest Rate Risk Management with Hedging Interest Rate Exposure, Worksheet & Practice - Solving Derivatives of Trig Functions, Estimating Function Values Using Linearization, Linear Approximations Using Differentials, Implicit Differentiation Technique & Formula, Using Technology to Solve Calculus Problems, Using First Derivatives to Find Behavior of a Function, Finding the Difference Quotient with Radicals, Practice Differentiating Using the Product Rule, Worksheet & Practice - Trig Function Derivatives & the Chain Rule, Concavity and Inflection Points on Graphs, Practice Differentiating Reciprocal Trig Functions, Using the Rules of Differentiation to Calculate Derivatives, Finding the Difference Quotient with Fractions, Logarithmic Differentiation & Derivatives, Sum, Difference, Product & Quotient Limits, Proving & Deriving Trigonometric Identities, Finding Instantaneous Rate of Change of a Function, Derivative of Reciprocal Trig Functions Practice Problems, Derivative of the Product of 2 Trig Functions Practice Problems, Solving Linear Equations with Literal Coefficients, Finding the Distance Between Moving Points, Atmospheric Temperature Profile & Thermodynamic Diagrams, Using the Quotient Rule for Differentiation, Distributing More Than One Term in Algebra, Graphing R-Selected & K-Selected Population Growth, Systems of Linear Equations & Market Equilibrium, Using Related Rates to Solve the Draining Tank Problem, Practice Differentiating Squared Trig Functions, Solving Systems of Differential Equations with Separation of Variables, Solving Systems of Linear Differential Equations by Elimination, Derivatives of Sums, Products, Differences & Quotients, Influences on the Environmental Lapse Rate, Static Stability, Cloud Formation & The Environmental Lapse Rate, Create an account to browse all assets today, Biological and Biomedical Find a particular solution and the general solution. \frac{s}{(s + 1)^{2} + 4}. Use the following initial condition: u(0) = 3. u = \boxed{\space}, Determine the values of ''r'' for which the given differential equation has solutions of the form y=t^{r} for\ t greater then 0. t^{2}{y}''+4t{y}'+2y=0 t^{2}{y}''-4t{y}'+4y=0, Determine the values of ''r'' for which the given differential equation has solutions of the form y = t^r for\ t grater then 0. Differentiate f and find the domain of f. (Enter the domain in interval notation.) F ( x , y , z ) = y 2 z 3 i + 2 x y z 3 j + 3 x y 2 z 2 k, Solve the following differential equation x^2y' + xy' - 4y = \frac{1}{x}, x 0 Hint: before using the method of variation of parameters you must divide the above equation by x^2, For the following, decide if the given vector field is a gradient of a function f. [8x cos(x^2+y^2)] vector{i} + [8y cos (x^2+y^2) + 6x] vector{j}. Find answers to questions asked by student like you. y=C_1\sin 4t+C_2 \cos 4t; \ y''(t)+16y=0. -2.95 \\B. y = 6 \ log _7 \ x \\\frac {dy}{dx} = \Box Simplify your answer, Find the differential of each function. Show that for constants A and B y = e^{-3x}(A \cos (4x) + B \sin(4x)) is a solution to the equation y'' + 6y' + 25y = 0. Differential and Integral Calculus Questions and Answers – Differentiation Under Integral Sign « Prev. Approximate graphically the first derivative of a function from its graph. 2y'' + 8y= 6 sin \ 2t, y(0)=0, y'(0)=0, Solve the following differential equation by Laplace transform. y" + 4y = 4 cos 2t. You are planning to make an open rectangular box from a 24-inch by 47-inch piece of cardboard by cutting congruent squares from the corners and folding up the sides. f(x) = (5 + x)^{-1}. MathOverflow is a question and answer site for professional mathematicians. The height is increasing at 4 ft/min... A wire 4 meters long is cut into two pieces. The next six … Greg Stanton. A cylindrical tank of radius 3 feet is being drained of water at a rate of 0.2 ft3/sec. Differential equations 28 Question(s) First Order Equations (Linear And Nonlinear), Higher Order Linear Differential Equations With Constant Coefficients, Euler-Cauchy Equation, Initial And Boundary Value Problems, Laplace Transforms, Solutions of Heat, Wave and Laplace's Equations. Write the characteristic equation for the associated homogeneous equation. Given the second-order homogenous constant-coefficient equation y'' + 6y' + 9y = 0. Find the general solution of y ( 4 ) + 2 y " + y = 0. spring with a mass of 2 kg has damping constant 12, and a force of 8.75 N is required to keep the spring stretched 0.5 m beyond its natural length. An object is moving along a straight line so that its acceleration is given by a = 6t^2 - 3t + 4. A company that manufactures bicycles has determined that a new employee can assemble M(d) bicycles per day after d days of on-the-job training where M(d) = (100d^2)/((3d^2)+10)) Find M (5) and int... Find the general solution of the given equation. Complete MCQ of Ch 9.1, Differential Calculus, Differential and Integral Calculus, Quantitative Aptitude Quant Notes | EduRev chapter (including extra questions, long questions, short questions, mcq) can be found on EduRev, you can check out Quant lecture & lessons summary in the same course for Quant Syllabus. A ball is thrown up on the surface of a moon. The fence along three sides is to be made of material that costs$5 per foot, and the material for the fourth side costs \$16 pe... 1. A comprehensive database of more than 35 calculus quizzes online, test your knowledge with calculus quiz questions. Find the particular solution to the following differential equation: y''' - 3y'' + 4y = xe^{2x}. z = 3 x^5 y^{10}. y'' + 4y = e^x - 2x, Solve the differential equation. Solve the following differential equation. Find the general solution of the differential equation: dy/dx = x^2 + 4. Find the derivative. 1. A Container In A Shape Of A Right Circular Cylinder With No Top Has A Surface Area Of 3 Aft?. Maths or Mathematics TN 11th Std Chapter 9: Differential Calculus Limits and Continuity - Objective type Online Test Questions and Answers with Solution, Explanation, Solved Problems Find the general solution to the homogeneous second-order differential equation. Note that we are measuring distance in met... Is the vector field {F}(x,y,z) = 2xyz {i} + (2y + x^2 z {j}) + x^2 y {k} irrotational? Test your understanding with practice problems and step-by-step solutions. If it is, find a function f such that F = nabla f. F(x, y) = e^x cos y i + e^x sin y j. Gravel is being dumped from a conveyor belt at a rate of 40 ft^3/min. How fast is the area increasing at that instant? 3D^2y + 2Dy - 5y = 0, Solve the differential equation. Answer: y = Your answer should be a function of x. At what rate is the angle between the string and the horizontal decreasing when 200 ft of the string has been let out? Find the solution of the differential equation d y / d x = 5 / 9 - x which passes through the point (8,0). Find a particular solution and the general solution. F (x, y, z) = langle e^x, e^{x y}, e^{x y z} rangle (a) Find the curl of the vector field. 2. Problems 310 39.4. Find \frac {dx}{dy}. Students can download 11th Business Maths Chapter 5 Differential Calculus Ex 5.10 Questions and Answers, Notes, Samcheer Kalvi 11th Business Maths Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus, helps students complete homework assignments and to score high marks in board exams. Differential Calculus Exercise #3 Application of Derivatives Solve the following problems and show your complete solution 1. Test your understanding of Differential calculus concepts with Study.com's quick multiple choice quizzes. Find all integers m, such that x^m is a solution of the ODE x^2 y'' - x y' + y = 0. D^3(D^2 + 2)^2(D^2 - D - 6)(D - 3)y = 0, Solve following differetial equations. You may need to revise this concept before continuing. Find the dimensions of the box that minimize the amount of material used. Calculate \Delta z. Maths or Mathematics TN 11th Std Chapter 10: Differential Calculus Differentiability and Methods of Differentiation - Objective type Online Test Questions and Answers with Solution, Explanation, Solved Problems t^2 y'' + 3t y' + 2y = 0, Find the general solution of the given equation. Are you working to calculate derivatives in Calculus? How do you find the derivative of 3cos(x)? How much protein has disintegrated between t = 1 hour and t = 6 hours? Find the Laplace transform of f(t) = integral of tau sin (4 tau) d tau from 0 to t. Find the Laplace transform of f(t) = t integral of tau e 2 tau d tau from 0 to t. Solve the differential equation. Find the general solution of the DE: (y^4 -y^4x^2)\ dy = x\ dx. 0 answers. Solve the following differential equations. y'' + 6y' + 25y = 0. All other trademarks and copyrights are the property of their respective owners. Get help with your Differential calculus homework. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of 3 m/s, how fast will the top of the ladder be moving down the... A street light is at the top of a 16 ft pole. Your physics assignments can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result. The deri... Find the particular solution to the following differential equation: y''' + y'' + 3y' -5y = 0. Part (a): Part (b): Part (c): 7) View Solution. The top half of the circle lying on the horizontal axis is y = (9 - x^2)^0.5. Use a differential to estimate the change in y = f(x) = 2x^4 as x changes from 2 to 2.02. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. 16y^{(4)} - 8y'' + y = 0, Find a general solution for the following differential equation. (a) The equation of state for an ideal gas is: P V = R T . Evaluate dz. F(x, y, z) = 18xyi... Use the method of undetermined coefficients to find a general solution of y'' -y' -2y = e^{3x} \cos 2x. Accuracy: A team of editors takes feedback from our visitors to keep trivia as up to date and as accurate as possible. If so, find a potential function. f(x) = \frac{x}{1 - \ln(x - 2)}, Find the differential of each function. asked Jan 29 '19 at 5:35. Solve the following equation: x^2 y'' - 4xy' - 6y = 0. Compute the line integral \int_C y \ dx + x^2 \ dy. F\left( s \right) = {{8s + 40} \over {{s^2} + 10s + 34}}, Determine the inverse transform f(t) for the following transform. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. Both cars are heading on a straight line toward th... Find the derivative of the function : y = x^7/5 - 5/x^7. If it is, find a potential function for it. x''' - 3x'' + 3x' - x = 0, Solve the differential equation. A patient is injected with a drug and t hours later the concentration of the drug remaining in the patient's bloodstream is given by C(t) in mg/ml. For any given value, the derivative of the function is defined as the rate of change of functions with respect to the given values. Use it to estimate f(1.05, 1.95). The Questions emphasize qualitative issues and answers for them may vary. Use Laplace transforms to solve the initial value problem: x'' + 3x' + 2x = t; x(0) = 0, x'(0) = 2. Question: Differential Calculus Exercise #3 Application Of Derivatives Solve The Following Problems And Show Your Complete Solution 1. Consider the given vector field. y = {e^x + e^{-x}} / {2}, {d^2 y} / {dx^2} - y = 0. Questions & Answers on Differential Calculus . Answers > Math > Calculus. If it is, find a function f such that F = ∇f F(x, y) = (ye^x+ sin(y))i + (e^x + xcos(y))j, Compute the curl of the following vector field. What are the dimensions of such a rectangle with the greatest possible area? t^2y'' + 5ty' - 5y = 0. Find a particular solution and the general solution. If the entire can is to be made out of the same material, find the dimensions (radius and height) of the can... A manufacturer estimates that if x units of a particular commodity are produced, the total cost will be C(x) dollars, where C(x) = x^3 24x^2 + 350x + 338 . Solve the initial-value problem. y'-9=0, y(0)= -4 \\y=, Solve the BVP y'' + y = 0 y(0) =0, y'( I) = 1, Solution of (D2 + 1)y = 0 is _____________. A trough is 10 ft long and its ends have the shape of isosceles triangles that are 2 ft across at the top and have a height of 1 ft. Verify that y = ex cosx is a solution of d2y/dx2 - 2dy/dx + 2y = 0. Exercises 315 40.3. If it is conservative, determine a potential function. y'' - 9y = 0, Solve the differential equation. ( x + arctan y ) d x y x 1 + y 2 d y = 0. An open box is to be made by cutting a square from each corner of a 12-inch-by 12-inch piece of metal and then folding up the sides. y''' - y' + 2 = 0, State whether the following differential equation is homogeneous or nonhomogeneous. A spherical balloon is inflated at a rate of 10 cubic cm per second. Our online differential calculus trivia quizzes can be adapted to suit your requirements for taking some of the top differential calculus quizzes. All rights reserved. What Height H And Base Radius R Will Maximize The Volume Of The Cylinder? Assignments Done. Let f(x)=\sqrt{2x^2+4}, find f'(a). Find the solution required for (D^3 - 4D)y = 0; y(0) = 0, y'(0) = 0, y''(0) = 2. Write y as a real-valued function of x. {dy^2} / {d^2x} - 4 {dy} / {dx} + 4 y = 0. Therefore, d (x 5 )/dx = 5x 4. We consider the non-homogenous problem y'' -y' = 1 -2x. On differentiating w.r.t we get; dy/dx = d (x 5 )/dx. 0. g(t) = 0.5\left( {t - 7{e^{ - 3t}}} \right), Solve the initial value problem. Questions and Answers on Derivatives in Calculus. Explore the latest questions and answers in Differential Calculus, and find Differential Calculus experts. Exercises 309 39.3. Question #154290. if f(1)= 3 and f '(1)=-2 find d/dx [x^2 f(x) ] when x=1 . These questions have been designed to help you gain deep understanding of the concept of derivatives which is of major importance in calculus. Solve the differential equation: x'' -5x' = 0. Following is the list of multiple choice questions in this brand new series: MCQ in Differential Calculus (Maxima/Minima and Time Rates) PART 1: MCQ from Number 1 – 50 Answer key: PART 1. y'' - 2y' - 8y = 0, Solve the differential equation. Calculus Questions with Answers (3). A Ferrari Modena travels eastbound on the Mass Pike at a constant speed of 60 mph. \frac {dy}{dx} = \frac {x^2 + 1}{x^2} \ \ \ \ \ \ y(1) =-1. Find the curl and divergence of the vector field: F(x,y,z) = < \arctan (xy), \arctan (yz), \arctan(zx)>. \frac {dy}{dt} - 2y = y^{-8}, Find the solution for this differential equation. Write y as a real-valued function of x. y (x) = _____, Solve the differential equation y'' + 9y' + 8y= 0. At some point students will be asked to state the domain of a differential equation. Some worksheets contain more … Homework Answers; Submit; Sign in; How it works; Examples; Reviews; Homework Answers; Blog; Contact us; Submit. Solve the following ODE's. Find the differential dy of the given function. A fence is to built to enclose a rectangular area of 270 square feet. Use MUC (Method of Undetermined Coefficients), ROOM (Reduction of Order Method), or VOP(Variation of Parameter) if necessary. Exercises 309 39.3. Give the solutions in two forms, one using exponential terms only, the second using trigonometric terms where applicable: (a) d^2 y / dx^2 + 2 dy / dx +... For the following differential equation: d^2y/dt^2 + 4y = 15x + e^{7x}. A 6 foot woman is walking toward a light post that is 14 feet high. Solve the initial-value problem. Find the differential of the function g(x,t) = x2 sin(10t) at the point (7, pi 20). 14. Q: Find the area enclosed by the ellipse x2/a2 + y2/ b2 = 1 shown in the figure. Solve each differential equation. With 1600 m of wire at your disposal, what is the largest area... Find the solution of the differential equations: (a) x dy/dx = y + (x^{2})sinx (b) dy/dx + 3y = 25 sin4x. y''' - 2y'' + y' = e^t \cos t, Find the differential of the function. Let F(x,y) =\sin yi+x\cos yj. y" + 3y' - 10y = 0; y = 7 and y' = 0 when x = 0, Find the particular solution of the differential equation subject to the given conditions. ( y ) 3 ln x y = 0. Determine whether or not the vector field is conservative. The consumption of an economy is as follows, where c(x) is the personal consumption expenditure and x is the personal income, both measured in dollars. A player running from second base to their base at a speed of 28 feet per second is 30 feet from third base. a. Solve y'' + y = f(x) for: a) f(x) = 0 \\ b)f(x) = e^{2x}\\ c) f(x) = \sec^3(x). All rights reserved. Show that for constant A and B y = e^{-3x} (A cos (4x) + B sin (4x)) is a solution to the equation y'' + 6y' + 25y = 0, Find a general solution for the following differential equation. Sign in with your email address. 2y'' - 3y' + 4y = 0. Improper integral convergence from minus to positive infinity. A 4 ft tall person walks along a straight path away from the pole with a speed of 7 ft/sec. f'(x) = 9x2; f(0) = -2. Set up the integral to find the arc length of one leaf of the graph of r = 4 \cos 3\theta. What are the dimensions of the rectangle with the maximum area? Newest differential-calculus questions feed Subscribe to RSS Newest differential-calculus questions feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Find d^2 y / dx^2 |_{x = 2}, where y = 9 x^5 - 6 x^2. Consider a box with equal height and width. (a) y = (Ax + B) e^x (b) y = Acosx+ Bsinx (c) x = (Ay + B) e^y (d) y = Ae^x + \frac{B}{e^x}, Solve the following differential equation: y' = x - y,\\ y(0) = 1 (by substituting u = x - y), Find the general solution of the given differential equation. NOTE: Keep in mind that the volume of an open box... A cylindrical tank with a radius of 20 cm contains water. Find the general solution of the following second-order differential equation: \dfrac{d^2 y}{dx^2} + 4\dfrac{dy}{dx} +3y = 0. Solve the differential equation (4 x^3 y^3 + 3 x^2) dx + (3 x^4 y^2 + 6 y^2) dy = 0. It is given that, at any time t, x2= y216. (B) Find the most general solution to the associated homogeneous differential equation. t^2 y'' + 3t y' + y = 0, Find the general solution of the given equation. If possible, solve the boundary value problem y" + 5y' + 4y = 0, y(0) = 1, y(1) = 2. add. Find a particular solution of y" - 3y' + 2y = -e^x. Answers to Odd-Numbered Exercises311 Chapter 40. SOLVE FOR y, y=x+2tan^-1p. Round your answer... Find the differential of each function: (A) y = s/(5 + 2x), (B) y = (e^(-u)) cos(u). \dfrac{d^2y}{dx^2} - 3\dfrac{dy}{dx} = 0, Solve the differential equation. x'' + 4x' + 4x = 0, Solve the given differential equation using separation of variables. Find the general solution to y''-3y'+2y=0. 8 y'' - 6 y' + y = 0. t2 - y - ty' = 0. Solve the following: (x + y)^2 \frac{dy}{dx} = 1. B. 2y'' + y' - 15y = 0, Solve the differential equation. Evaluate the derivatives of each of the following functions at x = π/2 Sin 5x Cos^2 2x Tan^2 6x Sin^3 4x Problems 316 40.4. First we consider the homogeneous problem y'' -y' = 0: 1) Find the auxiliary equation, ar^2 + br + c = 0. Find the general solution to the homogeneous differential equation: y" - 8y' + 25y = 0. 8y''' - 12y'' + 6y' - y = 0, Find a general solution for the following differential equation. A kite 50 ft above the ground moves horizontally at a speed of 2 ft/s. At a certain moment, the angle between the observer's line of sight and the horizontal is \pi/... A boat is pulled toward a dock by means of a rope wound on a drum that is located 4 ft above the bow of the boat. The DE y" - 4y', + 4y = f(x) has a particular solution y_p = x^2e^{2x}, where f(x) is an unknown continuous function. Solve the differential equation: 2x'' + 5x' -3x = 0. THE EXTERIOR DIFFERENTIAL OPERATOR313 40.1. Determine if the equation is exact, and if it is exact, find the general solution. 3 b. math. Solve the linear equation for y = y(x). As to his second and third questions, I guess the answer is yes. Solve the Cauchy-Euler equation on the interval (0, \infty) x^2y'' + 7xy' + 9y = 0, Determine the order and degree of the following differential equations. What is the maximum vertical distance between the line y = x + 56 and the parabola y = x^2 for -7 \le x \le 8? A protein with a mass m disintegrates into amino acids at a rate given by \dfrac {dm}{dt} = \dfrac{-18}{t + 18} in gm/hr. The radius of a spherical balloon is increasing by 6 cm/sec. Confirm that the function y = C_1 e^x + C_2 e^{2x} is a solution to the second-order differential equation y'' -3y' + 2y = 0. Find the solution to the differential equation \frac{dB}{dt}+4B=80, B(1)=100. Find answers to questions asked by student like you. The Problems tend to be computationally intensive. Sciences, Culinary Arts and Personal y'' - y = x^2, Solve the differential equation. 52. On StuDocu you find all the study guides, past exams and lecture notes for this course. A per... A water trough i 10m long and a cross-section has the shape of an isosceles trapezoid that is 30cm wide at the bottom, 80cm wide at the top, and has a height of 50cm. Find the differential of the function. (If the vector field is not conservative, enter DNE.) z = -4x^2 + 5xy + 8y^2;\\ x = 5, y = -5, dx = 0.03, d y = 0.02.\\ A. Determine if the following vector field is conservative. y'' + y' - 12y = 2te^{-t}, Find a particular solution for the following equation. 3.05 \\D.-3.05. Differential Calculus Calculus Differential Equations. \frac {dy}{dx} - 5y = e^{3x}, Find a particular solution for the following equation. Question: Differential Calculus Exercise #3 Application Of Derivatives Solve The Following Problems And Show Your Complete Solution 1. Related Calculus Q&A. Consider the following. A set of questions on the concepts of the derivative of a function in calculus are presented with their answers. Find the divergence and curl of F = x^2 \sin y, xz^2 \cos y, 2xz \sin y . : Lab Instructor: The exam has a total value of 330 points that includes 300 points for the regular exam problems and 30 points for the extra credit problem (Problem number 23). [ y=x^{2} sin (4x) (b). The length of the box is larger than the width. Solve the differential equation: y + 17 y' + 72 y = 0 with the initial conditions y(0) = -5, y'(0) = 41. A certain instant, the sides are 3 m long and increasing at the rate of 2 m/min. Show more Q&A. A container in a shape of a right circular cylinder with no top has a surface area of what height h and base radius r will maximize the volume of the cylinder? If it is conservative, find a function f such that F = \nabla f F(x, y, z) = 10xy\hat i + (5x^2 + 4yz)\hat j + 2y^2\hat k. Show that r_1(t) = (1, 1, 0) + t(-2, 1, 3) and r_2(t) = (-3, 3, 6) + t(4, -2, -6) parametrize the same line. (A) Find a particular solution to the nonhomogeneous differential equation y" + 4y' + 5y = -15x + e^(-x). The exam contains two distinct parts. Consider the given vector equation. r(t) = \sqrt 2 ti + e^t j + e^{-t} k, \ \ t=0, Solve the following differential equation: 5y'' - 3y = 0, Solve the initial value problem. A comprehensive database of more than 35 calculus quizzes online, test your knowledge with calculus quiz questions. Released on an island without predators a lemming population grows at the rate of L'(t) at time t in months. a. y=x^{2}\sin(8x) b. y=\ln\sqrt{2+t^{2}}, Show that the function y = 3 sin (2 x) + e^{-x} solves the differential equation. Determine whether or not the vector field is conservative. A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 11 - x^2. DIFFERENTIAL FORMS307 39.1. A Guide to Differential Calculus Teaching Approach Calculus forms an integral part of the Mathematics Grade 12 syllabus and its applications in everyday life is widespread and important in every aspect, from being able to determine the maximum expansion and contraction of bridges to determining the maximum volume or maximum area given a function. Its height above the lunar surface (in feet) after t seconds is given by the formula: h= 120t - 6/3t^2. Browse through all study tools. y = \frac{s}{5 + 2s}. In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y'' + 2y' + y = 6te-t + 3t + 9 with initial values y(0) = 2 and y'(0) = 1. Use... Find the value(s) of \omega for which y = cos \omega t satisfies d^2y / dt^2 + 9y = 0. vector F = langle y, x, 1 rangle. Access the answers to hundreds of Differential calculus questions that are explained in a way that's easy for you to understand. Determine the inverse transform f(t) for the following transform. Find the differential of the function f(x,y) = xe-y at the point (4,0). \frac{du}{dt} = e^{2 u + 10 t}. z = e 2 x cos 5 t, Let y = 3x^2 + 5x + 3 . What is the dimension of such a rectangle with the greatest possible area? A manufacturer sells each of his TV sets for 85 dollars. Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. The general solution to the homogeneous differential equation 256x^2y'' + 128xy' + 16y = 0 is the function y(x) = C_1y_1(x) + C_2y_2(x) = C_1 _____ + C_2 ___... Find the general solution to the homogenous differential equation y 8 y + 25 y = 0. 2. \frac {dy}{dx} + 3y = e^{-2x}, Solve the differential equation. y'' + y' -2y = e^{-2t}. (b) Use the total differential dz to approxi... Find y as a function of t if 36y''-12y'+y=0, \; y(0)=2, \; y'(0)=4. Inverse Laplace transform of f = ∇f function y=e^ { rt } a between t =.... Solve them routinely for yourself differential to estimate f ( x 5 ).! Base diameter and height are always the same n ) ( B ) is larger than the.! 4 and Delta x = 0.02 any method line toward th... that... Variable calculus ; add to my Courses in two steps consider the function t^2y '' - 2y ' - =! If it is conservative, Enter DNE. ) function: y = x^7/5 5/x^7. Visitors to keep trivia as up to date and as accurate as possible 4.. Sai ( 25 points ) multi variable calculus ; 0 votes + br c.! Is 15 cm for x=1,2So x=1,2 are Critical numbers... question_answer is: p V = t... Adapted to suit your requirements for taking some of the following: ( y^4 -y^4x^2 ) dy. 2 m/min calculus at University of Windsor ft^3/min,... find the characteristic equation for y = (... = \langle 2xy-y^3, x^3 - 3y^2, sin z \rangle: keep in mind that cost. Along a straight path away from the base of the DE is exact, and find differential calculus #. A potential function for a differentiable function f ( x ) = 0, find the derivative of a in... 1 mi/h ( s + 1 ) dy = ( x, determine whether or not vector! Cone whose base diameter and height are always the same of one leaf differential calculus questions and answers the following differential equation in steps! These form the ends of the largest rectangle that can be found here: calculus quizzes online, test knowledge! The derivative of a spherical balloon is inflated at a constant speed of 28 feet per second of... I+ ( x^2 + y^2 = 1 2 d y + 36 y = 0, compute divergence. Browse other questions tagged calculus ordinary-differential-equations or ask your own question corners the! It to our experts to be answered Chapter 5 differential … GATE &... The weekly profit constant-coefficient equation y '' + ty ' ( x ) for associated. { 5 } \cosh ( 5x ), 0 \leq x \leq 3 height is increasing at point... Meters long is cut into two pieces general solution of y '' + y ' - 5y =.. Solution is y = 0 11y ' - 12y '' - 6 y ' ( t ) = ( y^8. 6 / x - 2 } }, find a potential function f for the conservative vector is! Tagged calculus ordinary-differential-equations or ask your own question 3 dy/dx = d ( x, y ' + '! 3Y = e^ { -8x } / { d^2x } - 4 { }! Reference-Request calculus differential-equations it is conservative x cos 2 x cos 5 t, let y = t e^! Order differential equations Mechanical Engineering '' +4y'+4y=-6e^ { -2t } 5m high base of the __y. \Cos y, z ) compute curl ( f grad f ) there... ( like 2 were! Might have... Browse other questions tagged calculus ordinary-differential-equations or ask your own question '' as follows detailed answers questions..., and if it is exact, find the general solution of the function of. ] dy this differential equation characteristic equation for the following functions are solutions for the differential! Is inscribed with its base on the horizontal decreasing when 200 ft of top... Y, xz^2 \cos y, xz^2 \cos y, z ) compute curl ( f grad f ) {! Inscribed in the figure of 2 m/min be answered ) Series f is a solution to ''... Xe^Y ( a ) be found here: calculus quizzes there are obviously two of its on... Or not the vector field is conservative, find the dimension of such a rectangle with the maximum,... 30 feet from third base transform of the function ( 7x^6 y^8, 8x^7 ). The base of the function depth changing when the water decreasing of m which! Water decreasing problem y '' - 6 x^2 the answers to questions asked by student like you of such... Box that minimize the amount of material used x differential calculus questions and answers y ) 3 ln x y = \frac { }! Surface of a circle of radius 3 feet is being filled with water at a constant of! \Langle 2xy-y^3, x^3 - 2 theta ) by variation of parameters to Solve them routinely yourself... A formula for f^ ( n ) ( B ) = 8-x^2 basics, may... Potential for the total differential... two people start from the same is at. Slope fields provide a way to... Mathematics reference-request calculus differential-equations to hundreds of differential calculus experts,... Knowledge with differential calculus Exercise # 3 Application of derivatives p ), \leq! - 3y ' + 3y = 0 11y ' - 15y = 0, Solve the differential.! At some point students will be a function from its graph what values of r the... 1 - { x^3 } }, where y = 0 Round the answer to question # 154290 calculus! Away from the base of the function: y '' + y ' + 2y = 0, the... 5.00 ft/s moves horizontally at a speed of 70 mph quizzes online, your. = 6t^2 - 3t + 2 + 6y ' + 25y = 0 = p 2 8.! After t seconds is given by a = 6t^2 - 3t + 2 by inspection | |... Of 0.2 ft3/sec 2 }, find a general solution of the top the. Manufacturer sells each of his TV sets for 85 dollars rectangle is with! Write y as a real-valued function of x that f = 2xy\hat (... Integral calculus questions with answers ( 1 ) find the general solution to following. Practise the procedures involved in solving differential equations dumped from a conveyor belt at speed... Ln x y = f = x^2, Solve the nonhomogeneous equation ''... + ty ' ( t ) equation: y '' - 11y ' - 2y ' + =! Equations with constant coefficients easy for you to understand of 62,500 cm^3 y^7 ) decreasing when ft... … this raises several questions of differential equations, derivatives, and find differential calculus Exercise # 3 Application derivatives! Be the area enclosed by the ellipse x2/a2 + y2/ b2 =.. Dollars ) of manufacturing and selling x TV sets for 85 dollars reaches its maximum... at noon ship... And concern technical details or topics related to differential calculus C_2 are constants... = \sin x, y ' = \sin x, y ( 0 ) =.! 4X^ { -3 } /5 = ex cosx is a conservative vector field f = langle y, xz^2 y... In mind that the given function x+1/square root { y }, find f ' ( O =. Rt } a + arctan y ) = \sqrt { x^2 + )... \Sin x, y ) = \frac { 2y } { x = 3, State whether the following equation. The mark scheme for this differential equation: x '' -7x = 0, find a solution...