Concept explainers
Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function.
31. p(0) = 2,
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Calculus: Early Transcendentals (3rd Edition)
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Calculus, Single Variable: Early Transcendentals (3rd Edition)
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- Adding Functions A certain function f is the sum of two temperatures, one given by t2+3, and the other given by tt2+1. Find a formula for f in terms of t.arrow_forwardMinimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. Suppose g(x)=f(x) where f(x)0 for all x. Explain why the local minima and maxima of f and g occur at the same values of x. Let gx be the distance between the point 3,0 and the point (x,x2) on the graph of the parabola y=x2. Express g as a function of x. Find the minimum value of the function g that you found in part b. Use the principle described in part a to simplify your work.arrow_forwardA) If for a function f(x) the first and second derivatives at a point a are zero, is it possible that f(x) has a relative maximum at a? Reason your answer with an example. B) Study the continuity and asymptotic behavior of the function:arrow_forward
- 5) a function with a domain of (-2, infinity): Consider: Is the function continuous at x=1? If not, what type of discontinuity does the function have at x=1?arrow_forwardEven and odd functionsa. Suppose a nonconstant even function ƒ has a local minimum atc. Does ƒ have a local maximum or minimum at -c? Explain.(An even function satisfies ƒ(-x) = ƒ(x).)b. Suppose a nonconstant odd function ƒ has a local minimum atc. Does ƒ have a local maximum or minimum at -c? Explain.(An odd function satisfies ƒ(-x) = -ƒ(x).)arrow_forwardDomain,Interval of growth and decrease ,Asymptote by making the link with the characteristics of f(x) and g(x)arrow_forward
- True or false. if false, correct statement. if true, explain why. a. if a function, f, is continuous at a point c, then f is differentiable at the point c. b. if a function is concave down on its domain, then it will have a relative maximum. c. The derivative of a sum is the sum of its derivatives. d. the derivative of a function, f(x), is equal to limh->0 (f(x+h)-f(x))/h for all x values in the domain e. if f has an absolute minimum at c, then f'(c)=0arrow_forwardUsing rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using first two and then four rectangles. f(x)=1/x between x=2 and x=6 using two rectangles, the estimate for the area under the curve is ___. (round to three decimal places as needed) using four rectangles, the estimate for the area under the curve is ___. (round to three decimal places as needed)arrow_forwardq.no Use a graphing utility to graph the two functions f (x) = x2 + 1 and g(x) = lxl + 1 in the same viewing window. Use the zoom and trace features to analyze the graphs near the point (0, 1). Write a short paragraph describing the geometric significance of differentiability at a point?arrow_forward
- Approximate than the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, or relative minimum, an absolute maximum, an absolute minimum, or none of these at critical number on the interval shown. (Enter answers as a comma separated list)arrow_forward2(2) Show, and explain well your calculations. Please find: A) Image of the function given. B) Intervals where the g(x) function is positive or negative. C) Growth and decay intervals of the g(x) function. D) The intersection points of the affine function and its inverse.arrow_forwardlim 1/x^2 - 1 = infinity x--> 1+, M = 1000, find largest δ > 0arrow_forward
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