Preparation For Calculus Exam Answers

Introduction to calculus exam preparation problems and answers :

         Calculus is widely used in mathematics, science, and engineering. Calculus deals with limits, differentiation, and integration properties. Calculus is used for comparing the quantities of linear equations. Calculus mainly deals with integration and differentiation processes. It has many problems in differentiation and integration .  The quantities of  variables are recurrently changed by limits. For calculus exams we need to prepare integration and differentiation problems.

 

Calculus Exam preparation problems

 

Ex 1:

  Differentiate the given equation with respect to t. y = 5t4 - 2t3 + 8t2 + 46t.

Sol:

Given y = 5t4 - 2t3 + 8t2 + 46t.

Differentiating both sides, we have

      dy = (5 * 4)t(4 - 1) dt - (2 * 3)t(3 - 1) dt + (8 * 2)t(2 - 1) dt + 46t(1 - 1) dt.

           = 20t3 dt - 6t2 dt + 16t dt + 46 dt.

           = (20t3 - 6t2 + 16t + 46) dt.

dy / dt = 20t3 - 6t2 + 16t + 46.

Answer:

d/dt(5t4 - 2t3 + 8t2 + 46t) = 20t3 - 6t2 + 16t + 46.

Ex 2:

If t = x3 - 4x2 + 5. Find d2t/dx2.

Sol:

First step to find dt/dx. So, differentiate the given equation

Given t = x3 - 4x2 + 5.

        dt = 3x(3 - 1) dx - (4 * 2)x(2 - 1) dx + 0 dx.

            = 3x2 dx - 8x dx.

            = (3x2 - 8x) dx.

  dt / dx = 3x2 - 8x.

Then, once again differentiate the value of (dt/dx )with respect to x.

 we get  (3 * 2)x(2 - 1)  - 8x(1 - 1)

            = 6x dx - 8

            = (6x - 8)

Hence d2t/dx2 = 6x - 8.

Answer:

The second differentiation is  6x - 8.

Ex 3:

 

Integrate ∫ (3x2 - 4x4 + 3x) dx

Sol:

Given ∫ (3x2 - 4x4 + 3x) dx.

∫ (3x2 - 4x4 + 3x) dx = ∫ 3x2 dx - ∫ 4x4 dx + ∫ 3x dx.

                              = 3∫x2 dx - 4∫x4 dx + 3∫x dx.

                              = 3 (x3/3) - 4 (x5/5) + 3 (x2/2) + c.

                              = x3 - (4/5)x5 + (3/2)x2 + c.

∫ (3x2 - 4x4 + 3x) dx = x3 - (4/5)x5 + (3/2)x2 + c.

Answer:

∫ (3x2 - 4x4 + 3x) dx = x3 - (4/5)x5 + (3/2)x2 + c.