Concept explainers
EXPLORING CONCEPTS
Finding Functions Find differentiable functions f and g such that
Explain how you obtained your answers. (Note: There are many correct answers.)
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Calculus: Early Transcendental Functions
- Formula for Maximum and Minimum Values Find the maximum or minimum value of the function. f(x)=3x12x2arrow_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_forwardTrue 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_forward
- Finding a Pattern Develop a general rule for the nthderivative of xf (x), where f is a differentiable function of x.arrow_forwardlim x approaches infinity x1/xarrow_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_forward
- Curve sketching using limits and derivatives. Evaluate the function using algorithms and formulate conclusions. Use Desmos to verify the exercise.arrow_forwardlim. 3cosx x—> infinityarrow_forward84 . Think About lt Is it possible to find a differentiable function $f$ where $f(x)>0$ and $f^{\prime}(x)<0 ?$ If so, give an example. If not, explain why not.arrow_forward
- Proof! Let X bearandomvariable and let g(x) be a no0negative function. Then for r>0,0, P [g(X) ≥ r] ≤ Eg(X)/rarrow_forwardContinuity and limts: Show a function is continuous at aarrow_forwardFind differentiable functions f and g that satisfy the specified condition such that xlim5 f(x) = 0 and xlim5 g(x) = 0. Explain how you obtained your answersarrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning