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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**Extra resources for Concepts in Calculus I, Beta Version **

**Example text**

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.