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Solved: One-sided derivatives The left-hand and right-hand
Chapter 4, Problem 60AE(choose chapter or problem)
One-sided derivatives The left-hand and right-hand derivatives of a function at a point a are given by
\(f'_{+}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\) and \(f'_{-}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\)
provided these limits exist. The derivative f’(a) exists if and only if \(f'_{+}(a)=f'_{-}(a)\).
a. Sketch the following functions.
b. Compute \(f'_{+}(a)\) and \(f'_{-}(a)\) at the given point a.
c. Is f continuous at a? Is f differentiable at a?
\(f(x)=\left\{\begin{array}{ll} 4-x^{2} & \text { if } x \leq 1 \\ 2 x+1 & \text { if } x>1 \end{array} ; a=1\right.\)
Questions & Answers
QUESTION:
One-sided derivatives The left-hand and right-hand derivatives of a function at a point a are given by
\(f'_{+}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\) and \(f'_{-}(a)=\lim _{h \rightarrow 0^+}\ \frac{f(a+h)-f(a)}{h}\)
provided these limits exist. The derivative f’(a) exists if and only if \(f'_{+}(a)=f'_{-}(a)\).
a. Sketch the following functions.
b. Compute \(f'_{+}(a)\) and \(f'_{-}(a)\) at the given point a.
c. Is f continuous at a? Is f differentiable at a?
\(f(x)=\left\{\begin{array}{ll} 4-x^{2} & \text { if } x \leq 1 \\ 2 x+1 & \text { if } x>1 \end{array} ; a=1\right.\)
ANSWER:SOLUTION Given and a=1 STEP 1 (a). Sketch the following functions STEP 2 (b). Compute f +a)and f ()at the given point a. The left and the right derivatives of the function are f(a+h)f(a) f(a+h)f(a) f +a) = lim + h And f(a) = lim + h h0 h0 Provided a=1. Therefore 2 2 2 (4(1+h) )(41 ) (4(1+h +2h)4+1 h 2h f +1) = lim + h = lim + h = lim h = lim