Podcast
Questions and Answers
What is the equation of the function for which we are finding the tangent slope?
What is the equation of the function for which we are finding the tangent slope?
- f(x) = 5 + x
- f(x) = x^2 + 5 (correct)
- f(x) = x + 5
- f(x) = 5x + x^2
What is the required slope of the tangent line to the function at specific points?
What is the required slope of the tangent line to the function at specific points?
- 1/5 (correct)
- 1
- 0
- 5
Which of the following points is NOT a solution for the tangent line slope of 1/5?
Which of the following points is NOT a solution for the tangent line slope of 1/5?
- (5, 12)
- (10, 13) (correct)
- (0, 0)
- (−10, 2)
What conclusion can be made about the number of points where the slope of the tangent line is 1/5?
What conclusion can be made about the number of points where the slope of the tangent line is 1/5?
Which of the following pairs correctly represent the coordinates at which the tangent has a slope of 1/5?
Which of the following pairs correctly represent the coordinates at which the tangent has a slope of 1/5?
What value of x results in the function f(x) = x^2 + 5 yielding a slope of 1/5 for its tangent line?
What value of x results in the function f(x) = x^2 + 5 yielding a slope of 1/5 for its tangent line?
Which of the following pairs represents a point on the graph where the tangent slope is 1/5?
Which of the following pairs represents a point on the graph where the tangent slope is 1/5?
Which statement accurately reflects the existence of tangent points with a slope of 1/5?
Which statement accurately reflects the existence of tangent points with a slope of 1/5?
At which of the following points is the function f(x) = x^2 + 5 likely to have a slope of 1/5?
At which of the following points is the function f(x) = x^2 + 5 likely to have a slope of 1/5?
Which of the following options implies that there are no valid coordinates for the tangent slope of 1/5?
Which of the following options implies that there are no valid coordinates for the tangent slope of 1/5?
Flashcards
Tangent line slope
Tangent line slope
The slope of a line tangent to a function at a specific point.
Derivative
Derivative
The derivative of a function gives the slope of the tangent line at any point on the function's graph.
Finding the derivative
Finding the derivative
Finding the derivative of f(x) = x^2 + 5 using the power rule.
Solving for the x-coordinate
Solving for the x-coordinate
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Finding the y-coordinate
Finding the y-coordinate
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What does the derivative represent?
What does the derivative represent?
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How do you find the points where the tangent line has a specific slope?
How do you find the points where the tangent line has a specific slope?
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How to find the y-coordinate(s) for the points where the tangent line has the given slope.
How to find the y-coordinate(s) for the points where the tangent line has the given slope.
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What is the power rule?
What is the power rule?
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What is the x-coordinate of the point where the tangent line to f(x) = x^2 + 5 has a slope of 1/5?
What is the x-coordinate of the point where the tangent line to f(x) = x^2 + 5 has a slope of 1/5?
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Study Notes
Finding Tangent Line Slope Points
- The function is f(x) = x / (x + 5)
- The slope of the tangent line to f(x) at a given point is equal to the derivative of f(x) at that point.
- The problem asks for the x-values where the derivative (slope) is equal to 1/5.
- The correct answer is C. (0, 0) and (-10, 2).
Finding the Derivative (Slope)
- Calculating the derivative of f(x) using the quotient rule: f'(x) = [(x + 5)(1) - (x)(1)] / (x + 5)² f'(x) = 5 / (x + 5)²
Setting the Derivative Equal to 1/5
- Setting f'(x) = 1/5: 5 / (x + 5)² = 1/5
Solving for x
- Cross-multiplying to solve for x: 25 = x² + 10x + 25 x² + 10x = 0 x(x + 10) = 0 Solutions are x = 0 and x = -10
Finding Corresponding y-values
- Substitute x = 0 into the original function to find the corresponding y-value: f(0) = 0/(0 + 5) = 0. This gives coordinate (0, 0)
- Substitute x = -10 into the original function to find the corresponding y-value: f(-10) = (-10)/(-10 + 5) = (-10)/(-5) = 2. This gives coordinate (-10, 2)
Conclusion
- The points where the tangent line has a slope of 1/5 are (0, 0) and (-10, 2).
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