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Derivatives: Chain Rule

Please see the attached file for the fully formatted problems. For a composite function f(x) = g(u(x)) state the chain rule for the derivative f(x). For each of the following functions, compute the derivative, simplifying your answers. f(x)=ln(1 + x^2) f(x)= sin(x^2) f(x) = (sin x)^2 (a) For a composite function f(x)

Definition of derivative

Let f(x) be a continuous function of one variable. a) Give the definition of the derivative. b) Use this definition to find the derivative of f(x)=x^2+2x-5 c) Evaluate f'(2)

Dormitory Requirements for Students

Six students need to be placed in a dormitory. There are four double rooms, two single rooms, and two students cannot be placed together, how many ways are there to place the students?

Proof Regarding a Twice Differentiable Function

Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation. Question: Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0. Denote M = sup |f "(x)| where x is in [a,b]

Finding a rule for a sequence.

Find the rule for the sequence below. square 2x2 =10 3x3=40 +30=20 difference 4x4=90+50=20 difference 5x5=160+70=20 difference

Maximizing the area of a rectangle.

Find the dimensions of the rectangle of the largest area that has its base on the x axis and its other two vertices above the x axis and lying on the parabola y=8-x^2.

Polynomials and second derivatives

Find a polynomial p so that: p''(t)+3p'(t) + 2p(t) = (t^2)-2 for all numbers t. (note: p''= p double prime and t^2 = t raised to the power of 2)

Absolute Maximum and Minimum of a Given Function

Find the absolute maximum and absolute minimum values of f on the given interval. F(x) = sqrt(9-x^2) [-1, 2] or in other words: F(x) equals the square root of (9 minus x squared). The problem is also attached in MS word.

Using related rates to answer a pulley question.

Two carts A and B are connected by a rope 39 feet long that passes over pulley P. The point Q is on the floor directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/sec. How fast is cart B moving toward Q at the instant cart A is 5 feet from Q? Express solution using related rate n

Limits and derivatives

Find the derivative. a) f= 4-sqrt(x+3) b) f= (x+1)/(2-x) See attachment below for additional information.