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.
30. h(−1) = 2,
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Calculus, Single Variable: Early Transcendentals (3rd Edition)
- Values of related functions Suppose ƒ is differentiable on(- ∞, ∞) and assume it has a local extreme value at thepoint x = 2, where ƒ(2) = 0. Let g(x) = xƒ(x) + 1 and leth(x) = xƒ(x) + x + 1, for all values of x.a. Evaluate g(2), h(2), g'(2), and h'(2).b. Does either g or h have a local extreme value at x = 2?Explain.arrow_forwardlim f(x) as x approaches infinity lim f(x) as x approaches negative infinityarrow_forwardlimit as x approaches infinity of 3/(e^x-3) limit as x approaches negative infinity of 3/(e^x-3) How do I do this without using L'Hospital's rule?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_forwardh(x) = 4/((x-1)^2) Show from the definition that the function h(x) = 4/((x-1)^2) is increasing in the interval (-∞, 1) and decreasing in the interval (1,+∞).arrow_forwardCalculus - limit of the functions: Continuity How do I solve this in the image below?arrow_forward
- Composite Functions: What does (f*g^(-1))(x) and (g^(-1)*f)(x) stand mean, how do you solve them?arrow_forwardConsider the functions f(x) = x2 and g(x) = x3 . (a) Graph f and f′ on the same set of axes. (b) Graph g and g′ on the same set of axes. (c) Identify a pattern between f and g and their respective derivatives. Use the pattern to make a conjecture about h′(x) if h(x) = xn , where n is an integer and n ≥ 2. (d) Find f′(x) if f(x) = x4 . Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.arrow_forwardEnvironment of Forests Following the lead of the National Wildlife Federation, the Department of the Interior of a South American country began to record an index of environmental quality that measured progress and decline in the environmental quality of its forests. The index for the years 2004 through 2014 is approximated by the function where t = 0 corresponds to 2004. Find the intervals where the function I is increasing and the intervals where it is decreasing using interval notation.arrow_forward
- lim 1/x^2 - 1 = infinity x--> 1+, M = 1000, find largest δ > 0arrow_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_forwardConsidering the function f (x) = x ^ 4 - 32x ^ 2 - 144 (a) determine, if appropriate, the discontinuities of F (X) b) Determine whether f (x) is a function pair, odd or neither. c) Determine the intervals where f (x) is positive and where it is negative. Also determines the roots of the function. d) Determine the growth intervals and critical points of F (X) indicating the nature of these. e) Determine the intervals where the concavity is downwards and where it is up, as well as the points of inflection.arrow_forward
- College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning