1. Does this limit exist? If so, evaluate it. v2+x-V2-X lim x-&a...

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1. Does this limit exist? If so, evaluate it. v2+x-V2-X lim x->0 X 2. The definition of the derivative of a continuous function f(x) at the point x = a may be written as: f(x)-f(a) f'(a) lim = x->a x-a Let L be defined as: L x-a Assuming that exist, determine an expression for A such that L=Af'(a) Hint: Determine how to transform into the form in the definition of the derivative. f(x)-f(a) Vx-va 3. Consider the following function: x+1 + Is this function continuous over all values of x € (00, 00)? If not, over which intervals is it continuous? Determine p(x). Is this function differentiable over all values of X € (-00, x0)? If not, over which intervals is it differentiable? Find the following limits: lim f'(x) and lim f'(x) 4. Suppose a business sells X units of a product for a price each. The revenue R is R =xp Suppose it is found that the number sold (x) is the following function of price (p): 60,000 x = (0.9 22 1) If the price (p) is changing at the following rate: dp = $0.05/month dt What is the rate of change of revenue when the price is $4.00? 5. Find derivatives of the following functions: f(x) In(x) g(x) = In In(x)) h(x) Can you guess what the derivative of the following function is? j(x) = In( You are not required to derive this derivative, but you may confirm your guess by determining it if you wish. 6. 7. Use the linear approximation process to estimate the value of the function y = 2 sin(x) + 3cos(x) when X = 0.01 radians and when x = + 0.01 radians. In comparison to the actual values, what is the absolute and relative error for both? 8. Determine y' at the point ( 0, 1-) for: y + cos(y) = x2 - sin(x) Note that x and y are in units of radians.

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