(e) Use Euler's formula to prove the following formulas for cos(x) and sin(x) :eir + e-izeir – e-izcos(x) =and sin(r)22i(f) Determine all the number(s) c which satisfy the conclusion of Rolles Theorem forthe 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 ValueTheorem 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.1i. efii. 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) limsin(h)(c) lim r sin ()h40(b) lim1-cos(2)–(d) If u is a complex-valued function of a real variable, its indefinite integral ſ u(x)dxis an antiderivative of u. Evaluatee(1+i)= dx

Name: (e) Use Euler's formula to prove the following formulas for cos(x) an
Uploaded: 2024-03-20T06:47:22.000Z
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Description: (e) Use Euler's formula to prove the following formulas for cos(x) and sin(x) :eir + e-izeir – e-izcos(x) =and sin(r)22i(f) Determine all the number(s) c which satisfy the conclusion of Rolles Theorem forthe given function and interval.i. h(x) = x²