Final answer:
To find the slope of the line tangent to the given function at a specific point, you need to find the derivative of the function and evaluate it at that point. The instantaneous rate of change is equal to the derivative of the function. The equation of the line tangent to the function can be found using point-slope form.
Step-by-step explanation:
Summary:
To find the slope of the line tangent to the given function at a specific point, you need to find the derivative of the function and evaluate it at that point. The instantaneous rate of change is equal to the derivative of the function. The equation of the line tangent to the function can be found using point-slope form.
Step-by-step explanation:
(a) To find the slope of the line tangent to the function at a specific point, you need to find the derivative of the function and evaluate it at that point. The derivative represents the rate of change of the function at any point, which gives us the slope of the tangent line. For example, if the function is y = 2x^2, the derivative is dy/dx = 4x. To find the slope at x = 3, evaluate the derivative at x = 3: dy/dx = 4(3) = 12.
(b) The instantaneous rate of change of the function at a specific point is equal to the derivative of the function evaluated at that point. This represents the slope of the tangent line to the function at that point. Using the same example function as before (y = 2x^2), to find the instantaneous rate of change at x = 3, evaluate the derivative at x = 3: dy/dx = 4(3) = 12.
(c) To find the equation of the line tangent to the function at a specific point, you need to find the slope of the tangent line and the coordinates of the point. Once you have the slope and the coordinates, you can use the point-slope form of a linear equation to write the equation of the line. For example, if the slope is 3 and the point is (2, 5), the equation of the line is y - 5 = 3(x - 2).