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)
For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1. See attachment for diagram.
Using Cramer's Rule with a 3x3 system 3x+4y+z=17 2x+3y+2z=15 x+y =4
Given the function f(x) = e^(2x) a) Find the derivative b) Find the inverse (i.e. g(x)) c) Find the derivative of the inverse d) Find the value of g'(pi)
Find a real valued function such that its derivative exists in every point, but it is not continuous at least in one point.
The process of working with the definition of the derivative is shown using the example f(x)=3x-6.
The idea of the derivative is explained using the function x^2.
Use the (limit) definition of the derivative to find the derivative of f(x)=3x^2-2x+1?
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)
Find the first and second derivatives of the function y=((1-x)/x^2)^3
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?
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]
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
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.
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)
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.
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
Find the derivative. a) f= 4-sqrt(x+3) b) f= (x+1)/(2-x) See attachment below for additional information.