5. Jula found the derivative of some function and obtained the following: sinx f'(x) = 1+ cosx + 1+ cosx sinx Prove that the above derivative is the same as f'(x) = 2 csc(x)
og sinx cost D Check if the integral Sinx cosx dx is aboulte convergent, conditionally convergent or divergent let f«) contiewos, positive, decreasing function (0,1] such that SEGAOX convergent. Prove: lim xfx=0 on Xtot 5 let f be integrable function on [1]. foss x H XE [o,(Excluding some points). Prove that $x + 6) dx = }} Tip is for integrable, so is xf(x)
Q1. If g(x)=4cosx, then a) 4sinx b) -4 sinx c) -4cosx d) 4cosx Q2. The derivative of the function f(x)=-9x2 + 5x at x=-3 is a) -10 b) 41 c) 36 d) -49 x54 Q3. If g(x)= 3x +6x, *S*. then lim g(x)= =........ 8x+10, a) 48 b) 24 c) 72 d) 42 Q4. Find: lim (5+h)-25 Q5. Find: limano sin Q6. If y=x-*+cosx , then finden Q7. Using the definition, determine whether the function X57 is continuous at x=7....
PART 1A PART 1B Let f(t) = S4 0 < t < -4 < t < 27 and assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is even. Consider the following differential equation. dax 3 2 + 6x = f(t) dt Find a particular solution of the above differential equation of the form Xp(t) = Ë Ancos nat, a s(t,n) n=1 р n=1 and enter the function g(t, n) into the...
tion 5 sinx Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function h(x) 3int dt h'(x) = 3 cos(x)In(sinx) - h'(x) = 3 in (cos(x)s in(x))+ 1m3) - h'(x)=3\n(sin(x)cos(x)) - 31nYX h'(x) - 3sin(x) 31n(7) cosx 2 n'(x) = () - 3cos(x) 3inx sinx x h"(x) = 3sin(x)in(cosx) + 3inky 2x
Question 17 Find the derivative of the function f(x) = 4e*cosx . Simplify your answer. 0 a. 4e*(cosx-sinx) 0 b. 4e*(cosx + sinx) 0 C. 4e*cosx ( d. 4e*sinxcosx e. -4e* sinx A Moving to another question will save this response.
8. Find the solution to the differential equation y"+2y'+y=sinx using the method of undetermined coefficients. 1 COS X (a) y=ce' +ce' + -cosx 2 (b) y = ce' +cxe'+ (c) y = cxe' +cze cos x (d) y= c,e* + c xe" COSX 1 (e) y=ce' + ce + sinx 2 (f) y=ce' + exe* + sin x 2 (g) y=cxe' + e*- sinx 2 (h) y=ce' + cxe' 1 sinx 9. Use the method of undetermined coefficients to find...
3. Let I be the C-vector space with basis B = {1, cosx, sinx}. Define J: I + I by (Jf)(x) = 67 f(x +t) dt. Show that I is diagonalizable and find a basis of I consisting of eigenvectors of J.
6. Let g(t) = { 2te** t 20 6. Let g(t) be the probability density function of the continuous 0 t<0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(x = m) = { = 0.5. [7]
a) A step function has following characteristic 1for t > 0 -1 for t < 0 sgn(t) Use MATLAB plot this function over [-10, 10]. One display panel has two figures over two rows This is figure 1. It is displayed on top row of display panel. Give title; label x- and y- axes b) Plotting the Graph of a Sinc Function The sinc function is defined as sinx f(x) х whose value at x = 0 is sinx (sinx)...