4 The Concept of the Derivative

4.1 Limits

Practice Problems

Problem 4.1.1: Evaluate the following limit:

Problem 4.1.2: Evaluate the following limit:

Problem 4.1.3: Evaluate the following limit:

Problem 4.1.4: Evaluate the following limit:

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:

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: Estimating Limits from Graphs

Khan Academy: Estimating One-Sided Limits from Graphs

Khan Academy: Limits from Direct Substitution

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:

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

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

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

External Resources

Khan Academy: Formal Definition of a Derivative as a Limit

Khan Academy: Derivative of x2 using the Formal Definition

4.3 Continuity and Differentiability

Practice Problems

No practice problems for this section.