**Introduction to solve calculus exam questions:**

Calculus exams are mostly covers the differentiation and integration problems. Differentiation and integration methods are mostly used for solving the equations in calculus. Differential equation is also known as derivative of the equations. There are different topics are present in calculus. They are limits, infinite series, definite integral, continuity functions, and number series.

**Example 1:**

Find the rate of change of velocity for the given equation **v = 0.6t ^{2} + 4t^{3} - 12t.**

**Solution:**

Given v = 0.6t^{2} + 4t^{3} - 12t.

**Rate of change of velocity = dv/dt.**

Therefore, differentiate given equation with respect to t,

dv = (0.6 * 2)t^{(2 - 1)}dt + (4 * 3)t^{(3 - 1)} dt - 12t^{(1 - 1)} dt.

= 1.2t^{1}dt + 12t^{2}dt - 12t^{0}dt.

We know, **t ^{0} = 1.**

= 1.2t dt + 12t^{2}dt - 12.

= (1.2t + 12t^{2}- 12) dt.

dv/dt = 1.2t + 12t^{2} - 12.

**Answer:**

Rate of change of the velocity (dv/dt) = 1.2t + 12t^{2} - 12.

**Example 2:**

**If f(x) = 4x ^{2} / (12x + 6). Find f(4) and f(6).**

**Solution:**

Given f(x) = 4x^{2} / (12x + 6).

**Find f(4):**

Plug x = 4 in f(x)

f(4) = 4 * (4)^{2} / ((12 * 4) + 6).

= 4 * 16 / (48 + 6).

= 64 / 54.

Divide both numerator and denominator by 2,

**f(4) = 32/27.**

**Find f(6):**

Plug x = 6 in f(x),

f(6) = 4 * (6)2 / ((12 * 6) + 6).

= 4 * 36 / (72 + 6).

= 144 / 78.

Divide both the numerator and denominator by 6,

**f(6) = 24/13.**

**Answer:**

f(4) = 32/27.

f(6) = 24/13.

**Example 3:**

Find the value of **f '(5)**. **f(x) = 4x ^{2} + 3x - 1**.

**Solution:**

Given f(x) = 4x^{2} + 3x - 1.

f '(x) = d/dx (f(x)).

Differentiate given equation with respect to x,

f '(x) = (4 * 2)x(2 - 1) + 3x(1 - 1) - 0.

= 8x + 3.

**Find f '(5):**

Plug x = 5 in f '(x),

f '(5) = (8 * 5) + 3.

= 40 + 3.

= 43.

**Answer:**

f '(5) = 43.

1) Find f(3). f(x) = 12x^{2} - 3x + 7.

**Ans: 106.**

2) Find dy/dx.y = 5x^{3} + 2x^{2} - 4.

**Ans: 15x ^{2} + 4x.**

3) f(x) = 0.8x2 + 10x - 4. Find f '(2).

**Ans: 13.2.**