Contoh Soal & Jawaban Kalkulus

Materi:

1. Limit
2. Turunan
3. Persamaan Trigonometri
4. Aplikasi Turunan
5. Fungsi Dua Variable
6. Integral
7. Aplikasi Integral

1. Limit 

Contoh soal:

1. Lim
   x→3 (x^2-9)/x-3
   ⇒ ((3)^2-9)/(3-3)
   ⇒ (9-9)/(3-3)
   ⇒ 0/0
   Penyelesaiannya:
   ⇒ (x^3-9)/(x-3) == (x+3)(x-3)/(x-3)
   ⇒ x+3
   ⇒ 3+3 = 6
   
   
2. Lim
   x→-3 (x^2-9)/x-3
   ⇒ ((-3)^2-9)/(-3-3)
   ⇒ (9-9)/(-6)
   ⇒ 0/(-6) == 0
   atau
   ⇒ (x^3-9)/(x-3) == (x+3)(x-3)/(x-3)
   ⇒ x+3
   ⇒ -3+3 = 0
   
   
3. Lim
   x→0 (x^2-1)/x-1
   ⇒ (0^2-1)/(0-1)
   ⇒ (-1)/(-1) = 1
   
   
4. Lim
   x→1 (x^2-1)/x-1
   ⇒ (1^2-1)/(1-1)
   ⇒ 0/0
   penyelesaiannya:
   ⇒ (x+1)(x-1)/(x-1)
   ⇒ x+1
   ⇒ 1+1 = 2
   
   
5. Lim
   x→0 (sin 2x)/(sin x)
   ⇒ (sin 2(0))/(sin (0))
   ⇒ (sin 0)/(sin 0)
   ⇒ 0/0
   penyelesaiannya:
   (persamaan trigonometri)
   ⇒ (2 (sin x) (cos x))/(sin x)
   ⇒ (2 (cos x))
   ⇒ 2 (cos 0) == 2
   
   
6. Lim
   x→0 (cos 2x)/(sin x)
   ⇒ (cos 2(0))/(sin 0)
   ⇒ 1/0 == ~
   
   
7. Lim
   x→0 (sin x)/x^2
   ⇒ (sin 0)/(0^2)
   ⇒ 0/0
   penyelesaian:
   (diturunkan)
   ⇒ (cos x)/(2x)
   ⇒ (cos 0)/((2)(0))
   ⇒ 1/0 == ~
   
   
8. Lim
   x→0 2x/e^x
   ⇒ 2(0)/e^0
   ⇒ 0/1 == 0
   
   
9. Lim
   x→0 (sin x)/(cos x)
   ⇒ (sin 0)/(cos 0)
   ⇒ 0/1 = 0
   
10. Lim
    x→0 (cos x)/(sin x)
    ⇒ (cos 0)/(sin 0)
    ⇒ 1/0 = ~
    
11. Lim
    x→+~ (2x^2-x+1)/(x+1)
    (dibagi pangkat tertinggi)
    ⇒ (((2x^2)/(x^2))-(x/x^2)+(1/x^2)) / ((x/x^2)+(1/x^2))
    (catatan: k/~ mendekati 0, artinya sama dengan 0; k = konstanta)
    (coba bayangkan 1/10 = 0.1; 1/100 = 0.01; 1/100000000000 = 0.00000000001)
    ⇒ uraian: [2x^2/x^2 = 2],[x/x^2 = 1/x = 1/~ = 0], dst...
    ⇒ (2-0+0)/(0+0)
    ⇒ 2/0 = ~
    
12. Lim
    x→+~ (2x^2-x+1)/(x^2-1)
    (dibagi pangkat tertinggi)
    ⇒ ((2x^2/x^2)-(x/x^2)+(1/x^2)) / ((x^2/x^2)-(1/x^2))
    ⇒ (2-0+0)/(1-0)
    ⇒ 2/1 = 2
    
13. Lim
    t→~ (t^3-t^2)/(2t^3)
    ⇒ ((t^3/t^3)-(t^2/t^3)) / (2t^3/t^3)
    ⇒ (1-0)/2 = 1/2
    

2. Turunan 

Aturan dan Rumus dasar turunan

1. 1. d/dx (u ± v) = (du/dx) ± (dv/dy) = u' ± v'
   2. d/dx (u x v) = (u' v) + (u v')
   3. d/dx (u / v) = (u / v)' = ((u' v) - (u v')) / (v^2)
2. 1. f(x) = x^n
      f'(x) = n(x^(n-1))
      
   2. f(x) = sin x
      f'(x) = cos x
      
   3. f(x) = sin g(x)
      f'(x) = g'(x) cos g(x)
      
   4. f(x) = cos x
      f'(x) = -sin x
      
   5. f(x) = cos g(x)
      f'(x) = -g'(x) sin g(x)
      
   6. f(x) = e^x
      f'(x) = e^x
      
   7. f(x) = e^(g(x))
      f'(x) = g'(x) e(g(x))
      
   8. f(x) = ln x
      f'(x) = 1/x
      
   9. f(x) = ln g(x)
      f'(x) = g(x) / g'(x)
      

Contoh soal:

1. 1. f(x) = (x^2) (sin x)
      f'(x) =
      
   2. f(x) = (x^2) / (cos (2x))
      f'(x) =
      
   3. f(x) = e^(2x^2 + 1)
      f'(x) =
      
   4. f(x) = (cos (2x))/(sin (2x))
      f'(x) =
      
2. 1. f(x) = (2x^3 + 2x^2 + x + 1)
      f'(x) =
      
   2. f(x) = (4x^2 + x) / (2x^2 + 1)
      f'(x) =
      
   3. f(x) = (x^2 + x) / (sin (2x))
      f'(x) =
      
   4. f(x) = (sin x) / (cos x)
      f'(x) =
      
   5. f(x) = (sin (3x)) / (cos (3x))
      f'(x) =
      
   6. f(x) = (cos (3x)) / (sin (3x))
      f'(x) =
      
   7. f(x) = (e^(3x+1)) + (sin (e^2x))
      f'(x) =
      
   8. f(x) = (e^(3x^2)) / (ln (x^2 +1))
      f'(x) =
      
   9. S(t) = (t^3 - t^2 + 4t)
      S'(t) =
      
   10. S(t) = (sin (3t^2 + t)) / (cos (3t))
       S'(t) =
       
   11. S(t) = (e^2t + t) / (sin (2t))
       S'(t) =
       
   12. S(t) = (ln (4t^2 + t)) / (e^3t)
       S'(t) =
       

3. Persamaan Trigonometri

4. Aplikasi Turunan

5. Fungsi Dua Variable

6. Integral

7. Aplikasi Integral

To be continue...