(e) Use Euler's formula to prove the following formulas for cos(r) and sin(x) : eiz + e-iz eiz – e-iz cos(r) = and sin(r) = 2 2i

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(e) Use Euler's formula to prove the following formulas for cos(x) and sin(x) :
eir + e-iz
eir – e-iz
cos(x) =
and sin(r)
2
2i
(f) Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for
the given function and interval.
i. h(x) = x² – 2x – 8 on [–1,3]
ii. g(x) = r³ – 4x² + 3 on [0, 4]
(g) Determine all the number(s) c which satisfy the conclusion of the Mean Value
Theorem for the given function and interval.
i. h(x) = x³ – x² + x + 8 on [-3, 4] ii. P(x) = 4x³ – 82² + 7x – 2 on [2, 5]
(h) Write the number in the form a + bi.
1
i. ef
ii. ei
(i) Show that f(x) = 6x³ – 2x² + 4.x – 3 has exactly one real root.
(i) Show that f(r) = x" + 2x° + 3x³ + 14x + 1 has exactly one real root.
(k) Suppose we know that f(x) is continuous and differentiable on the interval [1,9],
that f(9) = 0 and that f'(x) > 8. What is the largest possible value of f(1)?
Transcribed Image Text:(e) Use Euler's formula to prove the following formulas for cos(x) and sin(x) : eir + e-iz eir – e-iz cos(x) = and sin(r) 2 2i (f) Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for the given function and interval. i. h(x) = x² – 2x – 8 on [–1,3] ii. g(x) = r³ – 4x² + 3 on [0, 4] (g) Determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval. i. h(x) = x³ – x² + x + 8 on [-3, 4] ii. P(x) = 4x³ – 82² + 7x – 2 on [2, 5] (h) Write the number in the form a + bi. 1 i. ef ii. ei (i) Show that f(x) = 6x³ – 2x² + 4.x – 3 has exactly one real root. (i) Show that f(r) = x" + 2x° + 3x³ + 14x + 1 has exactly one real root. (k) Suppose we know that f(x) is continuous and differentiable on the interval [1,9], that f(9) = 0 and that f'(x) > 8. What is the largest possible value of f(1)?
[5] Use series to evaluate the limits of the following:
(a) lim
sin(h)
(c) lim r sin ()
h40
(b) lim
1-cos(2)–
(d) If u is a complex-valued function of a real variable, its indefinite integral ſ u(x)dx
is an antiderivative of u. Evaluate
e(1+i)= dx
Transcribed Image Text:[5] Use series to evaluate the limits of the following: (a) lim sin(h) (c) lim r sin () h40 (b) lim 1-cos(2)– (d) If u is a complex-valued function of a real variable, its indefinite integral ſ u(x)dx is an antiderivative of u. Evaluate e(1+i)= dx
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