Chapter 2: Problem 21
In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=x^{2}-\frac{1}{2} \cos x $$
Chapter 2: Problem 21
In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=x^{2}-\frac{1}{2} \cos x $$
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Get started for freeElectricity The combined electrical resistance \(R\) of two resistors \(R_{1}\) and \(R_{2},\) connected in parallel, is given by $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$$ where \(R, R_{1},\) and \(R_{2}\) are measured in ohms. \(R_{1}\) and \(R_{2}\) are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is \(R\) changing when \(R_{1}=50\) ohms and \(R_{2}=75\) ohms?
Determining Differentiability In Exercises \(85-88\) , find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) $$ f(x)=\left\\{\begin{array}{ll}{(x-1)^{3},} & {x \leq 1} \\ {(x-1)^{2},} & {x>1}\end{array}\right. $$
Writing Use a graphing utility to graph the two functions \(f(x)=x^{2}+1\) and \(g(x)=|x|+1\) in the same viewing window. Use the zoom and trace features to analyze the graphs near the point \((0,1)\) . What do you observe? Which function is differentiable at this point? Write a short paragraph describing the geometric significance of differentiability at a point.
Proof. Prove the following differentiation rules. $$ \begin{array}{l}{\text { (a) } \frac{d}{d x}[\sec x]=\sec x \tan x} \\ {\text { (b) } \frac{d}{d x}[\csc x]=-\csc x \cot x} \\ {\text { (c) } \frac{d}{d x}[\cot x]=-\csc ^{2} x}\end{array} $$
A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 50 meters above the ground.
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