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Derivatives

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.

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:

Vectors : Product Rule, Gradient and Curl

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.

Speed of Plane/Boat

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

Derivatives and Limits

Find the derivative by the limit process. See attached file for full problem description. keywords: definition of the derivative, difference quotients

Applications of Derivatives and Rate of Change

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

Derivatives and Rate of Change

(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

Applications of Derivatives Word Problems : Maximizing Area and Revenue Functions

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

Applications of Derivatives : Maximum Volume and Tangents

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)

Real Analysis: Derivatives and Sequences

Suppose that f: [a,b] &#61664; R is differentiable, that 0 < m f '(x) M for x &#1108; [a,b], and that f(a) < 0 < f(b). Show that the equation f(x) = 0 has a unique root in [a,b]. Show also that for any given x1 &#1108; [a,b], the sequence (xn), xn+1 = xn - for n = 1, 2,..., is well defined (i.e. for each n, xn &#1108; [a

Force as the Gradient of Potential Energy

See attached file for full problem description. 1) Find the partial derivatives with respect to x, y, and z of the following functions: (a) f(x, y, z) = ax2 + bxy + cy2, (b) g(x, y, z) = sin(axyz2), (c) h(x, y, z) = aexy/z^2, where a, b, and c are constants. 2) Find the partial derivatives with respect to x, y, and z of t

Differentiation for a Tangent Line

Differentiation 1. Show that if the tangent to y=ekx at (a, eka) passes through the origin then a=1/k. 2. Find the value of a and b so that the line 2x +3y = a is tangent to the graph of f(x)=bx2 at the point where x = 3. See attached file for full problem description.

Derivatives, Differentiable Functions and Rate of Change

1. Functions f, g, and h are continuous and differentiable for all real numbers, and some of their values and values of their derivatives are given by the below table. x f (x) g(x) h(x) f'(x) g'(x) h'(x) 0 1 -1 -1 4 1 -3 1 0 3 0 2 3 6 2 3 2

Sunrise Baking Company markets dough nuts through a chain of food stores.

Sunrise Baking Company markets dough nuts through a chain of food stores. It has been experiencing overproduction and underproduction because of forecasting errors. The following data are its production in dozens of doughnuts for the past four weeks. Doughnuts are made for the following day; for example, Sunday's doughnut pro

Chain rule?

Please explain chain rule i'm a visual person please show several examples

Quotient and Composite Rules

1. a. use quotient rule to find derivative of this function. f(x) = (20+16x-x^2)/(4+x^2). b. Find any stationary points of the function from 1a. An use the first derivative test to see whether they are local maximum or local minimum of f(x). c. what are the maximum and minimum values of the function f(x) at interval [-6,2]

How fast is the brick falling after 2 seconds have passed?

A brick comes loose from near the top of a building and falls such that its distance s (in feet) from the street (after t seconds) is given by the equation s(t) = 200 - 16t^2 (see equation in attached file) How fast is the brick falling after 2 seconds have passed?

Finding Derivatives (12 Problems)

Answers and working to the questions: 1. Obtain dy for the following expressions. dx (a) y = (5x + 4)3 (b) y = (3 - 2x)5 (c) y = square root (5 - 0.6x) (d) y = (2 + 3x)-0.6 2. Differentiate the following with respect to o. (a) f(o) = sin(5o - 2) (b) f(o) = cos(4 - 3o) (c)

Derivatives and Rate of Change of Curves

See the attached file. 1: Both forms of the definitions of the derivative of a function f at number a. 2: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2ft/sec how fast will the foot of the ladder be moving away from the wall when the top is 5ft above the ground? 3: y':