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Let f(6) = -3 and f'(6) = 8. Then the equation of the tangent line to the graph of y = f(1) at z = 6 is !!

Let f(6) = -3 and f'(6) = 8. Then the equation of the tangent line to the graph of y = f(1) at z = 6 is !!

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Transcription for Educational Website:** --- ### Calculus Problem: Equation of a Tangent Line Given: - \( f(6) = -3 \) - \( f'(6) = 8 \) Then the equation of the tangent line to the graph of \( y = f(x) \) at \( x = 6 \) is --- *Note:* This problem involves finding the equation of the tangent line to a function at a given point using the point-slope form of the linear equation. **Steps:** 1. Identify the point on the graph: \((6, f(6))\) which is \((6, -3)\). 2. Identify the slope of the tangent line \( m \), which is \( f'(6) = 8 \). 3. Use the point-slope form of the equation \( y - y_1 = m(x - x_1) \): Plugging in the values: \[ y - (-3) = 8(x - 6) \] Simplify to: \[ y + 3 = 8(x - 6) \] Distribute and solve for \( y \): \[ y + 3 = 8x - 48 \] \[ y = 8x - 51 \] Thus, the equation of the tangent line is \( y = 8x - 51 \).
Transcribed Image Text:**Transcription for Educational Website:** --- ### Calculus Problem: Equation of a Tangent Line Given: - \( f(6) = -3 \) - \( f'(6) = 8 \) Then the equation of the tangent line to the graph of \( y = f(x) \) at \( x = 6 \) is --- *Note:* This problem involves finding the equation of the tangent line to a function at a given point using the point-slope form of the linear equation. **Steps:** 1. Identify the point on the graph: \((6, f(6))\) which is \((6, -3)\). 2. Identify the slope of the tangent line \( m \), which is \( f'(6) = 8 \). 3. Use the point-slope form of the equation \( y - y_1 = m(x - x_1) \): Plugging in the values: \[ y - (-3) = 8(x - 6) \] Simplify to: \[ y + 3 = 8(x - 6) \] Distribute and solve for \( y \): \[ y + 3 = 8x - 48 \] \[ y = 8x - 51 \] Thus, the equation of the tangent line is \( y = 8x - 51 \).
### Equation Input Field **Description:** The image depicts an input field for entering an equation, specifically for a variable `y`. Given the format, it appears to be part of a graphical calculator interface or an educational tool for plotting graphs or solving mathematical equations. **Details:** - The input field is labeled with `y =`, indicating this is where users can input their desired equation of `y` in terms of other variables (e.g., `y = mx + c`, `y = x^2 + 2x + 1`, etc.). - To the right of the input field, there is a button with a grid icon (three rows and three columns of small squares). This could be an indication to view or edit the graphing options, grid settings, or a menu for additional functionalities. This interface is likely used in teaching environments for students to practice entering and visualizing mathematical equations, helping them understand the relationship between algebraic expressions and their graphical representations.
Transcribed Image Text:### Equation Input Field **Description:** The image depicts an input field for entering an equation, specifically for a variable `y`. Given the format, it appears to be part of a graphical calculator interface or an educational tool for plotting graphs or solving mathematical equations. **Details:** - The input field is labeled with `y =`, indicating this is where users can input their desired equation of `y` in terms of other variables (e.g., `y = mx + c`, `y = x^2 + 2x + 1`, etc.). - To the right of the input field, there is a button with a grid icon (three rows and three columns of small squares). This could be an indication to view or edit the graphing options, grid settings, or a menu for additional functionalities. This interface is likely used in teaching environments for students to practice entering and visualizing mathematical equations, helping them understand the relationship between algebraic expressions and their graphical representations.
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