1) F(x, y, z) = xyz, denote the directional derivative of f at the point (x0, y0, z0) along the vector v by Lvf(x0, y0, z0). a. Find the gradient (Nabla)f(1, 2, 3) (congruent to) grad f(1,2,3) b. Find Lvf(1, 2, 3), where v = (-1, -2, 4) c. Find Luf(1, 2, 3), where u is the unit vector u = (2/3, -2/3, 1/3) d. Find the directi
Prove that if y = x/sqrt(x^2 + 2) then 3(y^2)(dy/dx) + (x)(d^2y/dx^2) = 0
Trying to "prove" the following product rule for vector derivatives given the functions: vec{A} = 2x x-hat + y y-hat + 4z z-hat and vec{B} = 2y x-hat - 3x y-hat for the following product rule: Del(A.B) [Gradient of A.B] = A X (curl B) + B X (curl A) + (A.Del)B + (B.Del)A where I denoted the gradient as Del.
If y = x / (sqrt (x^2 + 2)) Show that 3 y^2 dy/dx + x d2y/dx2 = 0
Locate the absolute extreme of the function on the closed curve. See attached file for full problem description.
See attached file for full problem description. 30. A boat is pulled into a dock... (a) Determine the speed of the boat when there is 13 feet of rope. What happens to the speed of the boat as it gets closer to the dock? (b) Determine the speed of the rope when there is 13 feet of rope. What happens to the speed of the ro
Find dy/dx by implicit differentiation. See attached file for full problem description.
See attached file for full problem description. 28. f(x) = (x3 + 3x + 2)/(x2 + 1) 32. f(x) = (cube root of x)(square root of x + 3) 14. f(x) = (x2 - 2x + 1)(x3 - 1) 18. (sinx)/x 20. f(x) = cosx/ex
See attached file for full problem description. 32. f(t) = 3 - 3/(5t) 12. y = t2 + 2t - 3 22. y = 5 + sinx 24. y = (3/4)ex + 2cosx
In Exercises 25-32, (a) find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point and (c) use the derivative feature of a graphing utility to confirm your results. 26. f(x) = x^2 + 2x + 1 ... (-3, 4) 30. f(x) = sqrt(x - 1) .
Find the derivative by the limit process. See attached file for full problem description. keywords: definition of the derivative, difference quotients
1.) (d/dx)(xe^(lnx^2))=? 2.) If x=e^(2t) and y=sin(2t), then (dy/dx)=? 3.) If y=xy+x^2+1, then when x=-1, (dy/dx) is ? 4.) A particle moves along the x-axis so that its acceleration at any time is a(t)=2t-7. If the initial velocity of the particle is 6, at what time t during the interval 0≤t≤4 is the particl
(1.) A particle moves along the x-axis so that at any time that t is greater than or equal to zero, its position is given by x(t)= t^3-12t+5. a.) Find the velocity of the particle at any time t. b.) Find the acceleration of the particle at any time t. c.) Find all values of t for which the particle is at rest. d.) Find the s
Use logarithmic differentiation to find dy/dx: y=(x+2)*sqrt[1-x^2]/(4*x^3)
Please see the attached file for the fully formatted problems.
Using the chain rule, show that if z = y + f(x^2 - y2), where f is differentiable, then partial derivatives y(dz/dx) + x(dz/dy) = x
Please see the attached file for the fully formatted problems.
Prove the following: If f(x) = 1 if x is greater than or equal to 0 and f(x) = 0 if x < 0, then there is no function F such that F'(x) = f(x) for every x in R. See #9 in the attached file.
1.) Mean Value Theorem: Let f(x)= x ln x a.) Write an equation for the secant line AB where A= (a,f(a)) and B= (b,f(b)) b.) Write an equation for the tangent line that is parallel to the secant line AB. 2.) Approximating functions: Let f be a function with f'(x)= sinx^2 and f(0)= -1 a.) Find the linearization of f at x=0
Use the chain rule to find dx/dy for the next 2 problems 1. y=5u^2+u-1; u = 3x+1 2. y=square root of (u); u = x^2+2x-4.
Find dy/dx by implicit differentiation 1. (3x^2+1)^4 for this problems find d^2y/dx^2 1. y/x - x/y = 5
Find the first and second derivative for for the next 2. z=2/(1+x^2) 3. f(x) = (x-1)/(x+1^2) find dy/dx by implicit differentiation 1. square root of (2x) + y^2 = 4
Compute the derivative of the function using the difference quotient and find the equation of the line that is tangent to its curve for x=c 1. f(x) = x^2-3x+2; c=1 2. y = (x+7)/(5-2x); c=1 Find the first and second derivative for for the next 3 problems 1. y=6x^5-4x^3-5x^2-1.
4. A Norman window consists of a rectangle with a semi-circle mounted on top (see the figure). What are the dimensions of the Norman window with the largest area and a fixed perimeter of P meters? 5. A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each pers
7. b) (fg)' = f'g + fg' c) (f/g)' = (f'g - fg') / g^2
Q: Find the derivative of the following function h(x) = x^5/(x^2-7). Please view Word Doc for a clearer version.
1 A box with its base in the xy-plane has its four upper vertices on the surface with equation z=48-3x^2-4y^2. What is the maximum possible volume. 2 Find the differential dw for w =ysin(x+z) 3 Find the equation of the plane tangent to z=-sin((pi)yx^2) at the point P =(1,1,0)
5. The number of new customers for an internet business, y, in a month is a function of the number of advertising email announcements, x, that are sent out that month. So y=f(x). a) What is the meaning of f(1250)= 22 and f'(1250)= 0.06? b) Use the information in part (a) to estimate f(1300) and f(1050). Of the two estimates, w
1 Find the point on the graph of y= e^x at which the curvature is the greatest. 2 Write the equation for the surface generated by revolving the curve x^2 - 2y^2 = 1 about the y-axis. Describe the surface 3 The parabola z = y^2, x =0 is rotated around the z-axis. Write a cylindrical-coordinate equation for the surface.
Investigating conic sections - you will be required to draw graphs from given information, describe features of graphs and find equations to fit given graphs.