# Itemized Textbook Solutions

## The Differentials: Example 7

Problem:Find $latex d\left[ln{\left(y^2\right)}+ln{\left(1+\sqrt y\right)}\right]&s=3&bg=ffffff&fg=000000$. SOLUTION: $latex =d\left[ln{\left(y^2\left(1+\sqrt y\right)\right)}\right]&s=1&bg=ffffff&fg=000000$$latex =d\left[ln{\left(y^2+y^\frac{5}{2}\right)}\right]&s=1&bg=ffffff&fg=000000$$latex =\frac{1}{y^2+y^\frac{5}{2}}\times\left(2y+\frac{5}{2}y^\frac{3}{2}\right)dy&s=1&bg=ffffff&fg=000000$$latex =\frac{y\left(2+\frac{5}{2}y^\frac{1}{2}\right)}{y^2\left(1+\sqrt y\right)}dy&s=1&bg=ffffff&fg=000000$$latex =\frac{2+\frac{5}{2}y^\frac{1}{2}}{y\left(1+\sqrt y\right)}dy&s=1&bg=ffffff&fg=000000$$latex =\frac{2+\frac{5}{2}\sqrt y}{y\left(1+\sqrt y\right)}dy&s=1&bg=ffffff&fg=000000$

## The Differentials: Example 6

d\left(cos{\theta}sin{\theta}\right)

## The Differentials: Example 5

d\left(cos{\theta}sin{\theta}\right)

## The Differentials: Example 4

d\left(cos{\theta}sin{\theta}\right)

## The Differentials: Example 3

Find d(1/sqrt(x^3+7)+4x^2-1).

## The Differentials: Example 2

Find dy if y=(2x)/(3x-1)

## The Differential

On your Calculus I, the notation $latex \frac{dy}{dx}&s=1&bg=ffffff&fg=000000$ was regarded as a single symbol to denote the derivative of a function y=f(x) with respect to x. More specifically, you have used the symbol $latex \frac{dy}{dx}&s=1&bg=ffffff&fg=000000$ to denote the limit of the quotient $latex \frac{\Delta y}{\Delta x}&s=1&bg=ffffff&fg=000000$ as $latex \Delta x&s=1&bg=ffffff&fg=000000$ approaches 0. This module introduces the concept of … Continue reading The Differential

## Normal Force Between Steel Piston and Steel Cylinder| Further Applications of Newton’s Laws: Friction, Drag, and Elasticity| College Physics| Problem 5.2

(a) When rebuilding her car’s engine, a physics major must exert 300 N of force to insert a dry steel piston into a steel cylinder. What is the normal force between the piston and cylinder? (b) What force would she have to exert if the steel parts were oiled?

## Friction Force Between Steel and Teflon| Further Applications of Newton’s Laws: Friction, Drag, and Elasticity| College Physics| Problem 5.1

A physics major is cooking breakfast when he notices that the frictional force between his steel spatula and his Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, he quickly calculates the normal force. What is it?