16. The derivative of f(x) = x + 4/x at x = 1 is 3. Choose the best approximatio
ID: 2882627 • Letter: 1
Question
16. The derivative of f(x) = x + 4/x at x = 1 is 3. Choose the best approximation to f(1 h) when h is very small.
(A) 5; (B) 5 3h; (C) 5 + 3h; (D) 5 h;
17. Suppose that f(2) = 2, f(2) = 0, f(2) = 2. Then x = 2 is
(A) local minimum; (B) local maximum; (C) inection point; (D) none of the above.
18. Suppose that the target value of the function f(x) is 1. Suppose that f(x0) = 0.97 and f(x0) = 3. What is the most reasonable adjustment to x0 to achieve the target value?
(A) increase x0 by 0.03; (B) decrease x0 by 0.03; (C) increase x0 by 0.01; (D) decrease x0 by 0.01
19. Suppose that the cost C = f(w), in dollars of buying a chemical is a function of the weight bought, w, in ounces. The statement f(192) = 0.4 means
(A) 1 pound of the chemical costs $19.20; (B) buying an extra ounce will cost you extra 40 cents; (C) paying an extra dollar will get you an extra 0.4 ounces of chemical; (D) 12 pounds of the chemical costs $0.4.
20. The number g(A) of goats that can be raised on a pasture at the same time is a function of the area A, in acres, of the pasture. What are the units of g(20)?
(A) goats; (B) acres; (C) goats/acre; (D) acre/goat
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Explanation / Answer
16) f(x)=x+ (4/x)
f(1)= 1+(4/1)=5
Given, f'(1)= -3 [Note:-- this can be found out by differentiating f(x) and substituting x as 1]
For finding f(1-h) we use linear approximation
f(x)= f(a) + f'(a) * Delta x [Delta x being (x-a)]
f(1-h)= f(1) + f'(1)* Delta x
= 5 + (-3*-h)
=3h + 5 (ANS)
17) For a point x, if f'(x) = 0 then it is a critical point.
When x is a critical point and----
(i) f"(x) >0, it is a local minima.
(ii) f"(x) <0, it is a local maxima.
In this case, f'(2)=0 [It is a critical point]
f"(2)<0, therefore 2 is a local maxima for the function.
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