Introduction to Calculus and Analysis - Richard Courant, Fritz John

## Problems for section 2.1

Find the derivative of the given function in Problems 1-21. $$f(x) = x^2$$ $$f(t) = t^2 + 3$$ $$f(x) = 1 - 2x^2$$ $$f(x) = 3x^2 + 2$$ $$f(t) = 4t$$ $$f(x) = 2 - 5x$$ $$f(t) = 4t^3$$ $$f(t)=-t^3$$ $$f(u) = 5\sqrt{u}$$ $$f(u) = \sqrt{u + 2}$$ $$g(x) = x\sqrt{x}$$ [Read More]

## 3.1 How to Set Up a Problem

In applications, a calculus problem is often presented verbally, and it is up to you to set up the problem in mathematical terms. The problem can usually be described mathematically by a list of equations and inequalities. The next two sections contain several examples that illustrate the process of setting up a problem. The examples in this section are from algebra and geometry, and those in the next section are from calculus. [Read More]

## 2.8 Implicit Functions

We now turn to the topic of implicit differentiation. We say that $$y$$ is an implicit function of $$x$$ if we are given an equation $$\sigma(x, y) = \tau(x, y)$$ which determines $$y$$ as a function of $$x$$. An example is $$x + xy = 2y$$. Implicit differentiation is a way of finding the derivative of $$y$$ without actually solving for $$y$$ as a function of $$x$$. Assume that $$dy/dx$$ exists. [Read More]

## 2.5 Transcendental Functions

The transcendental functions include the trigonometric functions $$\sin x,\ \cos x,\ \tan x$$, the exponential function $$e^x$$, and the natural logarithm function $$\ln x$$. These functions are developed in detail in Chapters 7 and 8. This section contains a brief discussion. 1 Trigonometric Functions The Greek letters $$\theta$$ (theta) and $$\phi$$ (phi) are often used for angles. In the calculus it is convenient to measure angles in radians instead of degrees. [Read More]

## 2.4 Inverse Functions

Two real functions $$f$$ and $$g$$ are called inverse functions if the two equations $$y = f(x), \ x = g(y)$$ have the same graphs in the $$(x, y)$$ plane. That is, a point $$(x, y)$$ is on the curve $$y = f(x)$$ if, and only if, it is on the curve $$x = g(y)$$. (In general, the graph of the equation $$x = g(y)$$ is different from the graph of $$y = g(x)$$, but is the same as the graph of $$y = f(x)$$; see Figure 2. [Read More]

## 2.3 Derivatives of Rational Functions

A term of the form $$a_1x + a_0$$ where $$a_1, a_0$$ are real numbers, is called a linear term in $$x$$; if $$a_1 \ne 0$$, it is also called polynomial of degree one in $$x$$. A term of the form $$a_2x^2 + a_1x + a_0, \ a_2 \ne 0$$ is called a polynomial of degree two in $$x$$, and, in general, a term of the form [Read More]

## 2.2 Differentials and Tangent Lines

Suppose we are given a curve $$y = f(x)$$ and at a point $$(a, b)$$ on the curve the slope $$f^\prime(a)$$ is defined. Then the tangent line to the curve at the point $$(a, b)$$, illustrated in Figure 2.2.1, is defined to be the straight line which passes through the point $$(a, b)$$ and has the same slope as the curve at $$x = a$$. Thus the tangent line is given by the equation [Read More]

## 2.1 Derivatives

We are now ready to explain what is meant by the slope of a curve or the velocity of a moving point. Consider a real function $$f$$ and a real number $$a$$ in the domain of $$f$$. When $$x$$ has value $$a$$, $$f(x)$$ has value $$f(a)$$. Now suppose the value of $$x$$ is changed from $$a$$ to a hyperreal number $$a + \Delta{x}$$ which is infinitely close to but not equal to $$a$$. [Read More]