Category: Examples

Problem: Find the integral: ∫(4×3) dx\int (4x^3) \, dx∫(4×3)dx

Solution: ∫4×3 dx=x4+C\int 4x^3 \, dx = x^4 + C∫4x3dx=x4+C

Answer = x4+Cx^4 + Cx4+C

Problem: The marks of 5 students are: 60, 70, 80, 90, 100. Find the mean.

Solution: Mean=Sum of marksNumber of students\text{Mean} = \frac{\text{Sum of marks}}{\text{Number of students}}Mean=Number of studentsSum of marks​ =60+70+80+90+1005=4005=80= \frac{60+70+80+90+100}{5} = \frac{400}{5} = 80=560+70+80+90+100​=5400​=80

Mean = 80

Problem: A right triangle has an angle of 30∘30^\circ30∘ and the hypotenuse is 10. Find the opposite side.

Solution: sin⁡(30∘)=oppositehypotenuse\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}sin(30∘)=hypotenuseopposite​ 12=opposite10\frac{1}{2} = \frac{\text{opposite}}{10}21​=10opposite​ opposite=5\text{opposite} = 5opposite=5

Opposite side = 5

Problem:
Find the derivative of: f(x)=x2+3xf(x) = x^2 + 3xf(x)=x2+3x

Solution: f′(x)=2x+3f'(x) = 2x + 3f′(x)=2x+3

Derivative = 2x+32x + 32x+3

Problem:
Find the area of a rectangle with length = 8 cm and width = 5 cm.

Solution: Area=Length×Width\text{Area} = \text{Length} \times \text{Width}Area=Length×Width Area=8×5=40 cm2\text{Area} = 8 \times 5 = 40 \, \text{cm}^2Area=8×5=40cm2

Area = 40 square centimeters.

Problem:
Solve for xxx: 2x+5=152x + 5 = 152x+5=15

Solution: 2x=15−5=102x = 15 – 5 = 102x=15−5=10 x=102=5x = \frac{10}{2} = 5x=210​=5

x=5x = 5x=5

Problem:
A basket has 12 apples. You eat 4 apples. How many are left?

Solution: 12−4=812 – 4 = 812−4=8

8 apples remain.