Calculating the reversal tangents of a function.
What is the equation of the reversal tangent of the following function? f(x)=-(x^3)+9x^2-(29x)+35
Derivatives
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Implicit differentiation : the chain rule and product rule.
Use implicit differentiation to find dy/dx if y^2 + 3xy + x^2 + 10 = 0 (1) where y is a function of the independent variable x.
First principles using standard theorems on trigonometric limits
Using the definition of the derivative and any standard limiting theorems, show that the derivative of (sinx)^2 is sin(2x).
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)
Equation of a straight line which is tangent to a curve
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
Using Cramer's Rule with a 3x3 system 3x+4y+z=17 2x+3y+2z=15 x+y =4
Working with inverse functions and their derivatives.
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 function whose derivative exists and is not continuous
Find a real valued function such that its derivative exists in every point, but it is not continuous at least in one point.
Show me how to use the definition of the derivative.
The process of working with the definition of the derivative is shown using the example f(x)=3x-6.
What am I doing when I take the derivative of a function?
The idea of the derivative is explained using the function x^2.
Find the derivative of a quadratic polynomial.
Use the (limit) definition of the derivative to find the derivative of f(x)=3x^2-2x+1?
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)
First and second derivatives
Find the first and second derivatives of the function y=((1-x)/x^2)^3
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.