微积分I（标准路径） · 免费预览

§1.3 The derivative of a function at a point

函数在给定区间上的平均变化率是如何定义的？这个量度量的是什么？

1. Introduction

Introduction

2. Reading the points behind a difference quotient

On the graph of a function $y = f(x)$ , two inputs $a$ and $a+h$ with $h \gt 0$ are marked on the $x$ -axis. Which statement about the points $(a, f(a))$ and $(a+h, f(a+h))$ and the segment joining them is correct?

3. The limit definition of the derivative

Which limit gives the derivative of a function $f$ at the input $a$ ?

4. What the derivative value measures

The computation $f'(a) = 1.5$ is reported for a function $y = f(x)$ . What does $1.5$ measure?

5. From secant slopes to a tangent slope

Difference quotients $\frac{f(2+h)-f(2)}{h}$ for $h = 0.1, 0.01, 0.001$ give $4.1, 4.01, 4.001$ . What value is $f'(2)$ ?

6. A difference quotient at a specific h

For $f(x) = x^2$ , compute the difference quotient $\frac{f(2.1) - f(2)}{2.1 - 2}$ exactly.

7. Simplifying a difference quotient

For $f(x) = x^2$ , simplify the difference quotient $\frac{f(2+h) - f(2)}{h}$ for $h \ne 0$ . The result is of the form $4 + h$ . Enter the constant term of that expression.

8. The derivative as a tangent slope

The derivative $f'(a)$ equals the slope of the tangent line to the graph of $y = f(x)$ at the point $(a, f(a))$ .

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