Get personalized answers to your specific questions with IDNLearn.com. Our community is ready to provide in-depth answers and practical solutions to any questions you may have.
Sagot :
Certainly! Let's break down the question and find the correct answer step-by-step.
1. Differentiability and Definition of the Derivative:
If [tex]\( f \)[/tex] is a differentiable function at [tex]\( a \in \text{Domain } f \)[/tex], it means that the derivative of [tex]\( f \)[/tex] at [tex]\( a \)[/tex], denoted [tex]\( f'(a) \)[/tex], exists.
2. Expression for the Derivative:
The expression given in the problem is:
[tex]\[ \frac{f(a + h) - f(a)}{h} \][/tex]
This expression is known as the difference quotient. It represents the average rate of change of the function [tex]\( f \)[/tex] over the interval from [tex]\( a \)[/tex] to [tex]\( a + h \)[/tex].
3. Limit Process to Define Derivative:
As [tex]\( h \)[/tex] approaches zero (from either the positive or negative direction), the difference quotient approaches the derivative [tex]\( f'(a) \)[/tex]:
[tex]\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \][/tex]
This limit, when it exists, gives the instantaneous rate of change of the function [tex]\( f \)[/tex] at [tex]\( a \)[/tex].
4. Interpreting the Derivative:
The derivative [tex]\( f'(a) \)[/tex] at a point [tex]\( (a, f(a)) \)[/tex] geometrically represents the gradient or slope of the tangent line to the graph of [tex]\( f \)[/tex] at that point.
5. Analyzing Each Choice:
- Choice A: The gradient of the normal line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is not correct since the normal line is perpendicular to the tangent line, and its gradient is related to but not the same as [tex]\( f'(a) \)[/tex].
- Choice B: The gradient of the secant line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is incorrect. The secant line connects two points on the graph, and its gradient is given by the difference quotient without taking the limit.
- Choice C: The gradient of the vertical line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is incorrect. A vertical line has an undefined gradient.
- Choice D: The gradient of the tangent line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is correct since the derivative [tex]\( f'(a) \)[/tex] gives the gradient of the tangent line at [tex]\( (a, f(a)) \)[/tex].
6. Conclusion:
Therefore, the expression [tex]\(\frac{f(a + h) - f(a)}{h}\)[/tex] as [tex]\( h \)[/tex] approaches 0 from both directions gives the derivative [tex]\( f'(a) \)[/tex], which is the gradient of the tangent line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex].
The correct answer is:
[tex]\[ \boxed{\text{D. The gradient of the tangent line to the graph of f at a point } ( a , f ( a ))} \][/tex]
1. Differentiability and Definition of the Derivative:
If [tex]\( f \)[/tex] is a differentiable function at [tex]\( a \in \text{Domain } f \)[/tex], it means that the derivative of [tex]\( f \)[/tex] at [tex]\( a \)[/tex], denoted [tex]\( f'(a) \)[/tex], exists.
2. Expression for the Derivative:
The expression given in the problem is:
[tex]\[ \frac{f(a + h) - f(a)}{h} \][/tex]
This expression is known as the difference quotient. It represents the average rate of change of the function [tex]\( f \)[/tex] over the interval from [tex]\( a \)[/tex] to [tex]\( a + h \)[/tex].
3. Limit Process to Define Derivative:
As [tex]\( h \)[/tex] approaches zero (from either the positive or negative direction), the difference quotient approaches the derivative [tex]\( f'(a) \)[/tex]:
[tex]\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \][/tex]
This limit, when it exists, gives the instantaneous rate of change of the function [tex]\( f \)[/tex] at [tex]\( a \)[/tex].
4. Interpreting the Derivative:
The derivative [tex]\( f'(a) \)[/tex] at a point [tex]\( (a, f(a)) \)[/tex] geometrically represents the gradient or slope of the tangent line to the graph of [tex]\( f \)[/tex] at that point.
5. Analyzing Each Choice:
- Choice A: The gradient of the normal line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is not correct since the normal line is perpendicular to the tangent line, and its gradient is related to but not the same as [tex]\( f'(a) \)[/tex].
- Choice B: The gradient of the secant line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is incorrect. The secant line connects two points on the graph, and its gradient is given by the difference quotient without taking the limit.
- Choice C: The gradient of the vertical line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is incorrect. A vertical line has an undefined gradient.
- Choice D: The gradient of the tangent line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is correct since the derivative [tex]\( f'(a) \)[/tex] gives the gradient of the tangent line at [tex]\( (a, f(a)) \)[/tex].
6. Conclusion:
Therefore, the expression [tex]\(\frac{f(a + h) - f(a)}{h}\)[/tex] as [tex]\( h \)[/tex] approaches 0 from both directions gives the derivative [tex]\( f'(a) \)[/tex], which is the gradient of the tangent line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex].
The correct answer is:
[tex]\[ \boxed{\text{D. The gradient of the tangent line to the graph of f at a point } ( a , f ( a ))} \][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Discover insightful answers at IDNLearn.com. We appreciate your visit and look forward to assisting you again.
