Textbook Question
Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.
2. Intro to Derivatives
Differentiability
2:33 minutesProblem 3.2.37
Textbook Question
One-Sided Derivatives
Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.
Verified step by step guidance1
Identify the piecewise function f(x) based on the graph: f(x) = x^2 for x < 0 and f(x) = x for x ≥ 0.
To find the left-hand derivative at P(0,0), compute the limit as x approaches 0 from the left: lim (x -> 0-) [f(x) - f(0)] / (x - 0). Since f(x) = x^2 for x < 0, this becomes lim (x -> 0-) [x^2 - 0] / x.
Simplify the left-hand limit expression: lim (x -> 0-) x^2 / x = lim (x -> 0-) x.
To find the right-hand derivative at P(0,0), compute the limit as x approaches 0 from the right: lim (x -> 0+) [f(x) - f(0)] / (x - 0). Since f(x) = x for x ≥ 0, this becomes lim (x -> 0+) [x - 0] / x.
Simplify the right-hand limit expression: lim (x -> 0+) x / x = lim (x -> 0+) 1.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-Sided Derivatives
One-sided derivatives are limits that assess the behavior of a function as it approaches a specific point from one side only. The right-hand derivative considers the limit as the input approaches the point from the right, while the left-hand derivative considers the limit from the left. If these two limits exist but are not equal, the function is not differentiable at that point.
Recommended video:
05:50
One-Sided Limits
Differentiability
A function is differentiable at a point if it has a defined derivative at that point, which means the function must be continuous and have a consistent slope from both sides. If the left-hand and right-hand derivatives at a point are not equal, the function is not differentiable there. This concept is crucial for understanding the smoothness and behavior of functions at specific points.
Recommended video:
05:53Finding Differentials
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for defining derivatives, as the derivative itself is the limit of the average rate of change of the function as the interval approaches zero. Understanding limits allows for the analysis of function behavior near points of interest, particularly in determining continuity and differentiability.
Recommended video:
05:50One-Sided Limits
5:02mWatch next
Master Determining Differentiability Graphically with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice