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# For which A does the derivative f′(0) exist and how much is it? (2) Let f (x

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CALCULUS PROBLEMS LIST 9

29.11.2011

(1) Let

f (x) =



ex2 − 1

cos x− 1 : x̸= 2kπ, k ∈ Z, A : x = 0.

For which A does the derivative f(0) exist and how much is it?

(2) Let

f (x) =



sin x− 1

cos2x : x /∈ {kπ + π2; k ∈ Z}, Ak : x = kπ +π2, k∈ Z.

For which Ak (k ∈ Z) do the derivatives f(kπ +π2)exist and how much are they?

(3) Let

f (x) =



x(x− 1)(x − 2)(x − 3)

sin(πx) : x /∈ Z, x2− 2x : x = Z.

Compute f(x) for those x ∈ Z, for which it exists.

(4) Let

f (x) =



e7x− 1

x : x̸= 0, 7 : x = 0.

Compute f(0). (5) Let

f (x) =



cos(πx) + 1

sin(πx) : x /∈ Z, x3− x : x∈ Z.

Compute f(x) for those x ∈ Z, for which it exists.

(6) Let

f (x) =



e3x− 3ex+ 2

x2 : ̸= 0,

A : x = 0.

For which A does the derivative f(0) exist and how much is it?

(7) Compute the derivative of order 3 of the function f, given by the formula:

(a) (x + 1)6, (b) x6 − 4x3+ 4, (c) 1 1− x, (d) x3log x, (e) e2x−1; (f) (x2+ 1)3, (g) ex2, (h) log(x2), (i) (x − 7)50.

1

(2)

(8) Derive the formula for the derivative of order n for the function f given by the formula:

(a) log(x10), (b) x log(x), (c) x, (d) sin2(x), (e) 1− x

1 + x, (f) xex, (g) sin(5x), (h) x7, (i) e4x, (j) x + 1

x, (k) x2e−x. (9) Prove that

(f · g)(n)(x) =

n k=0

(n k

)

f(k)(x)g(n−k)(x).

(10) Compute approximate values of the following expressions, using the three initial terms (zeroth, rst and second) of appropriately set up Taylor series. Estimate the error of the approximation using the Taylor's formula:

(a)

24, (b) 3

126, (c) 7 126, (d) sin(101), (e) arctan(101 ), (f)

50.

2

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