Be f(t) = (eat, e-at)andg(t) = (cos(at), sin(at)), with a > 0 check if a) f"(t) = a² f (t) and g"(t) = -a'g(t)b) f"(t) · f(t) > 0 and g"(t) · g(t) > 0

Q: Find f. f"(x) = cos(x), f(0) = 3, f'(0) = 6, f"(0) = 4

A:

Q: Find f. f(0) = = f"(0) = sin(0) + cos(8), f(0) = 3, f'(0) = 2

A: f"θ=sinθ+cosθ Integrating both sides  ∫f" θdθ=∫sinθdθ+∫cosθdθ f'θ=-cosθ+sinθ+C..... (i)  Put θ=0…

Q: Find f. f"(x) = cos(x), f(0) = 6, f'(0) = 4, f"(0) = 9 f(x) =

A: We can find the value of f(x) as below.

Q: Find f. f "(x) = 7 + cos(x), f(0) = -1, f(71/2) = 0 %3D %3D

A:

Q: Find f. f ''(t) = 4et + 6 sin t,    f(0) = 0,    f(?) = 0

A: To find f. f''(t)=4et+6sint,f(0)=0,f'(0)=0

Q: Let h(t) = tan(3t + 8). Then h'(5) is and h"(5) is

A: As we know that the chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1   Given: ht=tan3t+8 Now…

Q: Let h(y) = 20 cos y – 3 sin y. Then h'(y) =

A:

Q: Let h(x) = tan (2x + 6). Then h'(2) is and h'(2) is

A: topic- functions and derivatives

Q: Find f. f"(0) = sin(0) + cos(8), f(0) = 5, f'(0) = 2

A: Given, f"(θ)=sinθ+cosθ f(0)=5 and f'(0)=2

Q: If f(w) (3w), find f'(w). Use exact values. = COS f'(w) =

A: We can find the exact answer as below.

Q: nd r'(t), r"(t), and r'(t) · r"(t). r(t) = 4 cos(t)i + 3 sin(t)j a) r'(t) b) r"(t)

A:

Q: BE f(t) = (et, e¬at) G(t)= (cos(at), sin(at)), with a > 0. CHECK IF %3D a) F"(t) = a²F(t)ANDG" (t) =…

A: (a) If F(t)=eat,e-at and G(t)=cos(at),sin(at), then after differentiating F'(t)=ddt(eat),…

Q: Let f(z) = cos and g(z) = tan-(10z) Then if h(z) S(z) find: 9(z) h'(1) = N(즉) = |

A: The given functions are  f(x) = cos-1x/5 , g(x) = tan-110x   If, h(x) = f(x)/g(x)  Find h'(x) and…

Q: Find f. f"(e) = sin(e) + cos(8), f(0) = 2, f(0) = 2 f(e) =

A: Given problem:- f''(θ) = sin(θ) + cos(θ) , f(0) =2 , f'(0) =2  To find:- f(θ)

Q: If f(x)= cos(4x) Plot the function from (x=0) to (x=180)

A:

Q: xy"+y'+yx=ln x Solve the differential equation using the Parameter Variation Method

A: Given,xy"+y'+yx=ln xLety=xmy'=mxm-1y"=mm-1xm-2

Q: a fif f" (t) = 2e* +3 sin(t), f(0) =-8, ) =

A:

Q: Find f. f(0) = f"(0) = sin(0) + cos(8), cos(8), f(0) = 1, f '(0) = 3

A:

Q: If f(z)-6 tan , then find the value of f'(1) is 12 6 1 V2

A:

Q: Find f. f'(t) = sec(t)(sec(t) + tan(t)), = -1 f(t) =

A: Given that: f'(t)=sect(sect+tant),    -π2<t<π2, fπ4=-1

Q: If f(t) = sin−¹(3t), find ƒ'(t). Use exact values. ƒ'(t) =

A:

Q: Find f. f "(0) = sin 0 + cos 0, f(0) = 1, f'(0) = 3

A:

Q: b.9f Fex)=x Sin lgix), where gL1)= , and g' (1)= -1, then f'(1)=2

A:

Q: If f(z) = JE sin I, find f'(x).

A:

Q: Let f"(t) = 2t + sin t, f(0) = 1 and f'(0) = 2. Evaluate f(27).

A:

Q: Find f. f "(e) = sin(e) + cos(e), f(0) = 5, f (0) = 4 f(e)

A: Just use the basic integration of trigonometric and then find the value of integration constant from…