Vectors and Partial Derivatives: Laplace's Equation
Show that z = x^(2) - y^(2) + 2xy satisfies Laplace's Equation z (subscript)xx + z (subscript)yy.
Derivatives
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Vectors and Partial Derivatives
Verify: If w = tan (x^(2) + y^(2)) + x(y)^(1/2) , then w (subscript)xy = w (subscript)yx . keywords: differentiating, differentiate, differential
Vectors and Derivatives
^ ^ ^ Find k(t) for r(t) = t^(2) i + t^(3) j . keywords: differentiating, differentiate, differential
Velocity, Acceleration and Speed for an Arbitrary Time
^ ^ ^ Given r = 3 cos t i = 3 sin t j , find the velocity, acceleration and speed for an arbitrary time t.
Derivatives and Vector Functions
^ ^ Find r '(t) for r (t) = < ( 4 + 5t), (t - t^2) >.
Curve Sketching
See attached file for full problem description. 1.Boat A is sailing south at 24 km/h while boat B, which is 48 kilometres due south of A, is sailing east at 18km/h. a)At what rate are they approaching or separating one hour later? b)When do they cease to approach on another and how far apart are they at this time? c)What
Laws of Logarithms and derivative
Please see the attached file for full description 1.Evaluate the following logarithms. a)log22log55 b)log23200-2log210 2.Solve the following logarithmic equation. log25=log2(x+32)-log2x 3.Determine the derivative of each of the following functions. a)y=ln(x2+3x+4) b)y=logx2 c)y=lnx^2/(3x-4)^3 d)y=2x3e4x
Differential of a Function Solved.
Find the differential dy of the function y = 〖(1-3x^2)〗^(-2). Please use derivative to solve & show steps/work.
Derivatives : Demand and Cost Functions
The marketing research department for a computer company used a large city to test market their new product. They found that the demand equation was p= 1296-0.12 x^2. If the cost equation is C = 830+396 x, find the number of units that will produce maximum profit. I do not have a price per unit, only the above cost equatio
Thermodynamics Equation
Thermodynamics texts use the relationship (dy/dx)(dz/dy)(dx/dz) = -1 Prove that this equation is true. (Hint: Start with a relationship F(x,y,z)= 0 that defines x = f(y,z), y = g(x,z), and z = h(x,y) and differentiate implicitly.)
Derivatives : Related Rates and Rates of Change
1.Water is poured into a conical funnel at a rate of 1 cm3/s. The radius of the top of the funnel is 10 cm and the height of the funnel is 20 cm. Find the rate at which the water level is rising when it is 5 cm from the top of the funnel. I know that I am suppose to use the volume of the cone V=1pi r2h 3 2.A ligh
Derivatives and Rate of Change : Motion Problem
1.A body has an a equation of motion measured in metres after t seconds such that s=4t3-14t2+40t+8. 3 a) When and where is the body momentarily at rest? b) For what time interval is it moving forward? c) During what times is its acceleration negative? d) Draw three separate graphs for acceleration, velocity, and dis
Derivatives and Rates of Change
1.A ball is thrown directly upward from the ground. Its height above the ground is given by h=50t-5t2 Where h is measured in metres and t is measured in seconds. Determine a)its initial velocity b)the maximum height of the ball c)when the velocity is negative d)how long the ball is in the air e)the velocity with whi
analyzing the first and second derivatives.
1. Sketch the following function using intercepts and the information you get from analyzing the first and second derivatives. Mark all maximum, minimum and inflection points on the graph. Concentrate on showing the beahviour of the curve. Not on plotting points. Adjust the scales if necessary to make the curve easier to draw.
First and Second Derivatives
Please provide some assistance taking the first and second derivatives wrt B of this function. F(x)= Log ((2Y-1) xB) Calculate first and second derivatives for F(x). See the attached file.
Derivatives and Integrals Initial Voltages
1.) The current in a circuit is i = 2.00 cos 100t. Find the voltage across a 100-microfarad(uF) capacitor after 0.200 s, if the initial voltage is Zero (one microfarad(uF)=10 to the power of -6 F). 2.) Find the volume of the solid of revolution obtained by rotating the region bounded by y = cos x to the power 2, x=0, x = squa
Limits and Implicit Differentiation
Please show your work for the next 5 questions. 1. Lim (as x approaches 0) csc x (sec^2 x -1) 2. Use implicit differentiation to find an equation of the line tangent to the curve x^2 + y^2 at the point (3,-1) 3. Lim ( as x approaches infinity) of x^2+1 / square root of (x^4) + 1 4. 2 methods to find dy/dx from x^3 + y^3 =1
Derivatives and Rate of Change
1. The derivative of y= x lnx - x 2. The derivative of f(x)= log 2 X 3. Derivative of f(x)= e^x sin (x) 4. Derivative of f(x)= e^x/x^2 + 1 5. PV=RT a. P= pressure; v=volume; r=constant; t=temp At a certain time, the temp is maintained constant, the P=100ln/in^2 an
Finding a Derivative
Derivative of tan(1+cosx)
Description of Derivatives and Rate of Change
A beacon on a lighthouse 2000m away from the nearest point P on a straight shoreline revolves at the rate of 10 pi radians per minute. How fast is the beam of the light moving along the shoreline when it is 500m from P?
Derivatives : Product and Quotient Rules
1. For the following function find the value of the derivative at the specific point given using: - the definition of the derivative - the sum rule for derivatives Show that both methods lead to the same result. f(x)=-x3+3x2-2 at x=1 2. Find an equation to the tangent at the given point, using the Product Rule
Derivatives and Slopes of Tangents
1. Determine dy for each of the following relations. dx a) 9x2-16y2=1 b) y3+5xy+x3=1 2. Determine the slope of the curve 8x3+3xy+8y3=19 at the point (1,1). 3. Determine the equation of the tangent to the given curve at the given point. x2-y2-x=1 at (2,1) 4. Determine the equat
Derivatives
See attached file for full problem description. 1. Determine dy/dx for each of the following relations. a) 6x^2-3y^2=5 b) y^3+x^2-2x^2=0 2. Determine the slope of the curve 2x3+2y3-9xy=0 at the point (1,2). 3. Find dy for the relation 4x2+y2=16 using each of the following methods. i) Solve for y explicitly as a f
Derivatives
See attached file for full problem description. 1. Find dy for each of the following functions. dx a) y=3x^4-6x^2+2x b) y=3/x^2 c) y=(8x^4-5x^2-2)/4x^3 d) y=square root 5x - square root x/5 2. a) Find the slope of the tangent to the curve y=4x^3-3x^2+1 at the point where x=-1.
Derivatives, Tangent Lines and Rates of Change
1. A curve has the equation x²-4xy+y²=24 a) Show that dy/dx= (x-2y)/(2x-y) b) find the equation for the tangent to the curve at the point P (2, 10) The tangent to the curve at Q is parrallel to the tangent at P c) find the coordinates of Q 2. The diagram shows the coss-section of a vase. The volume of the water in
Exponential and Trigonometric Derivatives : Path of a Particle
Find the length of the path x(t) = e^t, y(t) = e^t cos t, z(t) = e^t sint, from 0 to 2. A particle moves along the path x = e^2 t, y = 2e^t, z = t. What distance is traveled between times t = 0 and t = 1?
Matrix of partial derivative
Jacobian matrix of partial derivative. See attached file for full problem description.
Polynomials, Fields and Derivatives
Determine whether the polynomials have multiple roots. See attached file for full problem description. 19. Let F be a field and let f(x) =...... The derivative, D(f(x)), of f(x) is defined by D(f(x)) = ...... where, as usual, ....... (n times). Note that D(f(x)) is again a polynomial with coefficients in F. The polynomi
Rule of Products
A bit string is a string of bits (0's and 1's). The length of a bit string is the number of bits in the string. An example, of a bit string of length four is 0010. An example, of a bit string of length five is 11010. Use the Rule of Products to determine the following: (a) How many bit strings are there of length
Gradients, Derivatives, Tangent Lines, Trajectory and Rates of Change
1) A particle is moving in R^3 so that at time t its position is r(t) = (6t, t^2,t^3). a. Find the equation of the tangent line to the particle's trajectory at the point r(1). b. The particle flies off on tangent at t0 = 2 and moves along the tangent line to its trajectory with the same velocity that it had at time 2. (Note: