By Marco Baronti, Filippo De Mari, Robertus van der Putten, Irene Venturi
This publication, meant as a realistic operating consultant for calculus scholars, contains 450 workouts. it truly is designed for undergraduate scholars in Engineering, arithmetic, Physics, or the other box the place rigorous calculus is required, and should enormously profit someone looking a problem-solving method of calculus. each one bankruptcy begins with a precis of the most definitions and effects, that is by means of a variety of solved workouts observed by means of short, illustrative reviews. a range of issues of indicated options rounds out every one chapter.
A ultimate bankruptcy explores difficulties that aren't designed with a unmarried factor in brain yet in its place demand the mix of a number of thoughts, rounding out the book’s coverage.
Though the book’s basic concentration is on features of 1 actual variable, simple traditional differential equations (separation of variables, linear first order and incessant coefficients ODEs) also are mentioned. the fabric is taken from genuine written checks which have been brought on the Engineering tuition of the collage of Genoa. actually millions of scholars have labored on those difficulties, making sure their real-world applicability.
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Extra resources for Calculus Problems
Therefore f is strictly increasing, hence invertible. Take now y ≥ 1. The point x ≥ 1 satisfies f (x) = y provided that 2 elog x = y = elog y =⇒ log2 x = log y =⇒ log x = √ log y =⇒ x = e log y . √ Evidently log y ≥ 0 and hence its exponential is in [1, +∞). Therefore√the image of f is [1, +∞) and f −1 : [1, +∞) → [1, +∞) is given by f −1 (y) = e log y . This exercise is standard and simply requires to see that the given function actually maps the set [1, +∞) bijectively onto itself. For injectivity, it is immediately seen that f is increasing.
Draw the graphs of the functions fe and fo . 14 Consider the function f : R → R defined by: ⎧ log x − x 3 ⎪ ⎪ ⎪ x>0 ⎪ ⎨ 2 f (x) = 0 x=0 ⎪ ⎪ ⎪ log(−x) − x 3 ⎪ ⎩ x < 0. 2 Draw the graphs of the functions fe and fo . 15 Consider the function f : R → R defined by: f (x) = αx 3 + βx 2 + γ x + δ α, β, γ , δ ∈ R. Find α, β, γ , δ ∈ R in such a way that: (a) f is even; (b) f is odd; (c) f (1) = f (−1) = 0 and f (2) = f (−2) = 1. 16 Let g : [−1, 3] → R be the function whose graph is in depicted in Fig. 46.
4y In this exercise, the basic idea is again to view g as a composite function. Once this is done, then finding the domain and the image is achieved by carefully following what each map ( f and h) does. Most of the effort actually goes into finding the image of g. Inversion is done by inverting each of f and h. 5 Consider the function f (x) = |x| + x 2 − 1. (a) Establish if −5/4 ∈ Im( f ). (b) Find the largest neighborhood of x0 = 1 on which the function is invertible, and write the explicit analytic expression of the inverse.