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- “Interpretation of Derivative” sketch the graph of a function that satisfies the given conditions.y = f(x) a b The graph of f', the derivative of f, is shown in the figure above. Which of the following describes all relative extrema of f on the open interval (a,b)? (A) One relative maximum and two relative minima (B) Two relative maxima and one relative minimum (C) Three relative maxima and one relative minimum (D) One relative maximum and three relative minima (E) Three relative maxima and two relative minimaExample 2: Sketching the function from the derivative Use the features of each differentiate function f(x)to sketch the graph of its original function f(x). y = f'(x) RHS Mathematics Depart f(x)=y=0.lx'-0.2x² -1.9x+2 =0.1(x+4Xx-1Xx-5) y=f(x) y=f(x)
- Calculating derivatives at a point Let h(x) = ƒ(g(x)). Use the values to calculate h′(1) and h′(2).Differentiation of Piecewise Functions Describe how you would differentiate a piecewise function. Use your approach to find the first and second derivatives of f(x) = x|x|. Explain why f"(0) does not exist.Comparing graphs of f and f': Use the given graph to estimate the value of each derivative.
- Function Point g(t) = −5 cos t + 1 (π, 6) Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your result.Using the definition of derivative of a function at a point, f′(a)=lim x→a f(x)−f(a)/x−a, find the slope of the curve at the given point or state that the slope is undefined. f(x)=13x^2 + x - 14; a=-3 The slope is ______ Note: type 'undefined' if you think the slope of the curve is undefined at the given points.QI: In Leibnitz theorem, If f and g are functions of a variable x, then nth derivative of (f.g) can be defined as: (f. g)(n) = f(n)g + "c,f (n-1)(g)(1) + "c,f (n-2)( Where (f.g)(n) = D"(f.g), D" - g)(2) + ... + "cif(1)(g)(n-1) + f. g(m) d" and "c, n! %3D %3D (n-r)! r! Prove that p(n) is true for all positive integral value of n.
- Behavior of P'(a) In a carefully-worded sentence, describe the behavior of P'(a) as a increases in value. 80000 60000 40000 What does your observation about the behavior of the derivative P' as a increases in value tell you about the behavior of the function P? 20000 2Calculating derivatives at a point Let h(x) = f(g(x)). Use the valuesin Table 3.2 to calculate h′(1) and h′(2).4- 3+ 2+ 1+ X- 1 3 4 Graph of f' The graph of f', the derivative of f, is shown in the figure above. If f(0) = 20 which of the following could be the value of f(5)? (A) 15 (B) 20 (С) 25 (D) 35 (E) 40