Concepts in Calculus I, Beta Version by Miklos Bona, Sergei Shabanov

By Miklos Bona, Sergei Shabanov

From the college of Florida division of arithmetic, this can be the 1st quantity in a 3 quantity presentation of calculus from a innovations standpoint. The emphasis is on studying the recommendations at the back of the theories, now not the rote crowning glory of difficulties.

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15. Derivatives of Trigonometric Functions In this section, we show how to compute the derivatives of trigonometric functions. First, we compute (sin x) . This will be a somewhat lengthy procedure, due to the fact that this is the first trigonometric function we will differentiate and we will have to apply new methods. However, once we know the derivatives of sin x and cos x, it will be much simpler to deduce the derivatives of other trigonometric functions, since those functions can be obtained from sin and cos, and then the various differentiation rules can be used.

5. Let f (x) = 3x3 − 4x2 + x − 2 + 4ex . Compute f (x). Could it happen that f and g are two different functions, but f (x) = g (x) for all x? (12) Could it happen that f and g are two different functions, f and g are two different functions, but f (x) = g (x) for all x? 14. 1. The Product Rule. We mentioned in the last section that, in gen- eral, (f g) = f g . For instance, if f (x) = 2x + 1 and g(x) = x + 2, then (f g)(x) = 2x2 + 5x + 2, so (f g) (x) = (2x2 + 5x + 2) = 4x + 5, while f (x) = 2 and g (x) = 1, so f (x)g (x) = 2.

1) (2) (3) (4) (5) 2 Find limx→3 x −4x+3 . x−3 x2 −4x+7 Does limx→3 x−3 exist? Find limx→0 cos x. x2 . Find limx→0 |x| Let f (x) = x be equal to the largest integer that is at most as large as x. 7) = 3. Note that f is often called the floor function or integer part function. Find the values a for 8. LIMIT LAWS (6) (7) (8) (9) (10) (11) (12) 31 which limx→a f (x) exists. If a is such that f has no two-sided limits at a, decide if f has one-sided limits at a. Let g(x) = x be equal to the smallest integer that is at least as large as x.

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