Remarks (W.H.) If f(x) and g(x) differentiable functions of x, and a, b are constants then dla f(x) + b g(x)] = a df(x) + b dx dg(x) are dx Or (a f(x) + b g(x))' = a f'(x) + b g'(x)
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- Consider a differentiable function f with domain R and derivativesf'(x)=-aebx(1+bx) and f"(x)=-abebx(2+bx) , with a and b nonzero real numbers.The function has only one critical point x=-1/b and a local maximum at x=-1/bUse the Second Derivative test to find the value(s) of a and b5. Consider the function f(x) = |x| + 2 at x = 0. Provide a table of inputs and outputs that demonstrate the limitdefinition of the derivative, numericallyWe consider the rational function f(x) = p(x)/ q(x) = 2x ^2 + cx − 4c / x^2 − x − 2 . Here c is an unknown constant. If the limit L = limx→2 f(x) is a finite number, then what is c?
- 1. Find the absolute maximum and minimum values of f(x)=10x(2-lnx) on [1,e^2] 2. Find the limil. Uisng L'Hopital's Rule : lim x arrow 0 sin^2(3x)/4xIf F admits continuous partial derivatives and the equation F ( 3x - y, z2 - x2)= 0 defines z implicitly as a function of x and y, then taking u = 3x - y and w = z2 - 2x then the value of the expression image1 corresponds to image 21. A critical number of f′(x) is an inflection point of f(x).True or False 2. If the radius of a circle is increasing at a constant rate, then so is the circumference.True or False 3. If two variables x and y are functions of t and are related by the equationy = 1 −x^2, then dy/dt = −2x.True or False 4. When finding the global minimum of a function on an interval, you must usethe first or second derivative test to verify that your point is a global minimum.True or False
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- f(x) = (x2-x-1) e-x is restricted to the domain x ≥ 0 and it has two critical points. One at x = 0 and the other at x = 3 At what value x does f(x) attain its maximum?2. Suppose that f(x) is a function continuous for every value of x whose first derivative is f'(x) = 2(1-x)/1+x^2 and f"(x)= 4x(x^2-3)/ (1+x^2)^2 Further, assume that it is known that f has a horizontal asymptote at y = 0. and a. Determine all critical points of f.a.) The stationary points of f are at x = -98, x = -37, and x = 25. Moreover, f' (-100) < 0,f'(-73) < 0, f' (30) < 0. Classify each stationary point as either a relative minimum, relative maximum, or neither. b.) Find the absolute maximum and absolute minimum of f (x) = 4x^3 + 3x^2 - 6x +1 on [0,3] (Hint: the derivative of f is f'(x) = 6(2x-1)(x+1) .)