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

3. Techniques of Differentiation

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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2:04 minutes

Problem 63a

Textbook Question

Let f(x) = x² - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 0.

Verified step by step guidance

Step 1: To find the values of x for which the slope of the curve y = f(x) is 0, we need to find the derivative of f(x) with respect to x. The derivative, f'(x), represents the slope of the curve at any point x.

Step 2: Differentiate f(x) = x^2 - 6x + 5. Using the power rule, the derivative of x^2 is 2x, and the derivative of -6x is -6. The derivative of a constant, 5, is 0.

Step 3: Combine the derivatives to get f'(x) = 2x - 6.

Step 4: Set the derivative equal to 0 to find the x-values where the slope is 0. So, solve the equation 2x - 6 = 0.

Step 5: Solve for x by adding 6 to both sides and then dividing by 2. This will give you the x-value(s) where the slope of the curve is 0.

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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 measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve at any given point. For the function f(x), finding the derivative f'(x) will allow us to determine where the slope of the curve is zero.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection on the graph of the function. In this problem, we will find the critical points by setting the derivative of f(x) equal to zero to identify where the slope of the curve is flat.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the properties of quadratic functions, including their vertex and axis of symmetry, is essential for analyzing the behavior of f(x) in this problem.