Determine the slope of the tangent to the curve [tex]y=x^2[/tex] at the point with [tex]x[/tex]-coordinate 11.
A. 22
B. 1
C. 33
D. 11
Answer (2)
Find the derivative of the function: d x d y = 2 x .
Evaluate the derivative at x = 11 : 2 ( 11 ) = 22 .
The slope of the tangent to the curve y = x 2 at x = 11 is 22 .
Explanation
Problem Analysis We are given the curve y = x 2 and asked to find the slope of the tangent at the point where x = 11 . The slope of the tangent to a curve at a given point is given by the derivative of the function evaluated at that point.
Finding the Derivative First, we need to find the derivative of the function y = x 2 with respect to x . Using the power rule, we have d x d y = 2 x
Evaluating the Derivative Next, we evaluate the derivative at x = 11 to find the slope of the tangent at that point: d x d y ∣ x = 11 = 2 ( 11 ) = 22
Final Answer Therefore, the slope of the tangent to the curve y = x 2 at the point where x = 11 is 22.
Examples
Understanding the slope of a tangent line has many real-world applications. For example, in physics, if y represents the distance an object has traveled and x represents time, then the slope of the tangent line at a particular time gives the object's instantaneous velocity at that time. Similarly, in economics, if y represents the cost of production and x represents the number of units produced, then the slope of the tangent line at a particular production level gives the marginal cost of production at that level.
The slope of the tangent to the curve y = x 2 at x = 11 is 22, which is found by evaluating the derivative d x d y = 2 x at that point. Therefore, the correct option is A. 22.
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