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Derivatives

Finding the Derivatives in Calculus

Q#8. For f(x)=2-5X^2, Find: i) f`(x). Answers: a. -5x b. -10x c. -10x d. 0 Q#9. Consider f(x)=3-4*Square root(X) i) Find f`(x) Answers: a. - square root of X / 2 b. - 4 / square root of X c. -2/square root of x d. -8 / Square root of x. Q#10. Consider f(x)=3x/x+9 i) find f`(x) Q#7. A per

Derivatives : Velocity and Displacement

27. A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10 m/s. It strikes the ground with a speed of 60 m/s. How tall is the building? Answer: y0 = 178.57 m 35. A stone is dropped from rest at an initial height h above the surface of the Earth. Show that the speed wit

Derivatives and Rate of Change

Early one morning it began to snow at a constant rate. At 7 AM a snowplow set off to clear a road. By 8 AM it had traveled 2 miles but it took two more hours for the snowplow to go another 2 miles. Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour), at what time did it start to snow?

Derivatives, Rate of Change and the Fundamental Theorem of Calculus

1. The resale value of a certain industrial machine decreases at a rate that depends on its age. When the machine is t years old, the rate at which its value is changing is -960e^-t/5 dollars per year. a) Express the value of the machine in terms of its age and initial value. b) If the machine was originally worth $5,200,

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

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 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

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

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

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:

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

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

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