# 4 The Concept of the Derivative

## 4.1 Limits

### Practice Problems

**Problem 4.1.1: **Evaluate the following limit:

Answer: 39

**Problem 4.1.2: **Evaluate the following limit:

Answer: 3

**Problem 4.1.3: **Evaluate the following limit:

Answer: 5

**Problem 4.1.4: **Evaluate the following limit:

Answer: 4/7

**Problem 4.1.5: **Evaluate the following limit:

Answer: DNE (negative infinity), DNE (positive infinity), DNE

**Problem 4.1.6: **Evaluate the following limit:

Answer: DNE (positive infinity), DNE (positive infinity), DNE (positive infinity)

**Problem 4.1.7:** Evaluate the following limit:

Answer: DNE

**Problem 4.1.8:** Use the figure below to answer the following questions.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

Answer: 4, -2, DNE, 4; 2, -2, DNE, 1; 0.8, 0.8, 0.8, 0.8

### External Resources

Khan Academy: Intro to Limits

Khan Academy: Estimating Limits from Graphs

Khan Academy: Estimating One-Sided Limits from Graphs

Khan Academy: Limits from Direct Substitution

Khan Academy: Limits from Factoring

## 4.2 Rates of change and the difference quotient

### Practice Problems

**Problem 4.2.1: **Find the derivative of the following function using the **difference quotient**:

Answer: Derivative is

**Problem 4.2.2: **Find the derivative of the following function using the **difference quotient**:

Answer: Derivative is

**Problem 4.2.3: **Find the derivative of the following function using the **difference quotient**:

Answer: Derivative is

**Problem 4.2.4: **Find the derivative of the following function using the **difference quotient**:

Answer: Derivative is

### External Resources

Khan Academy: Formal Definition of a Derivative as a Limit

Khan Academy: Derivative of x^{2} using the Formal Definition

## 4.3 Continuity and Differentiability

### Practice Problems

No practice problems for this section.

### External Resources

Khan Academy: Continuity Introduction

Khan Academy: Worked Continuity Example

Khan Academy: Differentiability and Continuity