Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Tangent Lines and Derivatives

Calculus

2. Intro to Derivatives

Tangent Lines and Derivatives

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2:20 minutes

Problem 3.2.2

Textbook Question

If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.

Verified step by step guidance

Step 1: Understand that the derivative of a function, f′(x), represents the slope of the tangent line to the curve y=f(x) at any point x.

Step 2: Recognize that you are given f′(x) = 3x + 2, which is the expression for the slope of the tangent line at any point x on the curve.

Step 3: To find the slope of the tangent line at x = 1, substitute x = 1 into the derivative: f′(1) = 3(1) + 2.

Step 4: To find the slope of the tangent line at x = 2, substitute x = 2 into the derivative: f′(2) = 3(2) + 2.

Step 5: To find the slope of the tangent line at x = 3, substitute x = 3 into the derivative: f′(3) = 3(3) + 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is represented as f'(x) and provides the slope of the tangent line to the curve at that specific point. In this question, f'(x) = 3x + 2 indicates how the slope varies with x.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of this line is equal to the derivative of the function at that point. To find the slope of the tangent line at x = 1, 2, and 3, we evaluate the derivative at these x-values.

Evaluating Functions

Evaluating a function involves substituting a specific value into the function to find its output. In this context, we will substitute x = 1, 2, and 3 into the derivative f'(x) = 3x + 2 to find the slopes of the tangent lines at these points. This process is essential for determining how the function behaves at specific locations.

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Related Practice

Textbook Question

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y= √x; P(4, 2)

Textbook Question

y = 2/x; P(1, 2)

Textbook Question

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x)=x²+1; Q(3, 6)

Textbook Question

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x) = 1/x; Q (-2, 4)

Textbook Question

The line tangent to the graph of f at x=5 is y = 1/10x-2. Find d/dx (4f(x)) |x+5

Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/ √x; a= 1/4

Textbook Question

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = 1/ √x; a= 1/4

Textbook Question