Calculus 1 Final Exam Review

In this article, we will review the final exam of Calculus 1 in detail and chapterwise. The syllabus of Calculus 1 includes Limit, Continuity, Derivative, Integration, and their applications.

Limit Final Exam Review

Question 1: Find $\lim\limits_{x \to 2} \dfrac{x^2-5x+6}{x^2-4}$.

Solution:

At first, we will factorize both the numerator and the denominator. Note that x²-5x+6 = x²-3x-2x+6 = x(x-3)-2(x-3) = (x-2)(x-3).

And x²-4 = (x-2)(x+2) by the formula a²-b² = (a-b)(a+b).

$\therefore \dfrac{x^2-5x+6}{x^2-4}$ $=\dfrac{(x-2)(x-3)}{(x-2)(x+2)}$ $=\dfrac{x-3}{x+2}$

Thus, the limit will be equal to

$\lim\limits_{x \to 2} \dfrac{x-3}{x+2}$

= $\dfrac{2-3}{2+2}=-\dfrac{1}{4}$.

Limit: Definition, Formulas, Examples

Continuity Final Exam Review

Question 2: Find the value of c if the following function f(x) is continuous at x=1.

$f(x)=\begin{cases}cx-3 \quad \quad \text{ if } x\geq 1 \\ x^2+2cx \quad \quad \text{ if } x<1 \end{cases}$

Solution:

As f(x) is continuous at x=1, we must have that

lim_x→1+ f(x) = lim_x→1- f(x) = f(1).

Therefore, [cx-3]_x=1 = [x²+2cx]_x=1

⇒ c⋅1-3 = 1²+2c⋅1

⇒ c-3 = 1+2c

⇒ 2c-c = -3-1 ⇒ c=-4.

Thus, if the value of c is -4 then the function f(x) will be continuous at x=1.

Must Read:

Continuity of a Function: Definition, Properties, Solved Examples

Discontinuity of a Function: Definition, Types, Examples

Derivative Final Exam Review

Question 3: Find the derivative of f(x) = x(x-1)(x-2).

Solution:

We have x(x-1)(x-2) = x(x²-x-2x+2) = x(x²-3x+2) = x³-3x²+2x.

So $\dfrac{d}{dx}\{x(x-1)(x-2)\}$

= $\dfrac{d}{dx}(x^3-3x^2+2x)$

= $\dfrac{d}{dx}(x^3)-3\dfrac{d}{dx}(x^2)+2\dfrac{d}{dx}(x)$

= 3x²-6x+2 by the power rule of derivative

Also Read:

Derivative: Definition, Formulas, Examples

Integration Final Exam Review

Question 4: Find the integral ∫ (√x+x³)dx

Solution:

We have ∫ (√x+x³)dx

= ∫x^1/2dx + ∫x³dx

= $\dfrac{x^{1/2+1}}{\frac{1}{2}+1}+\dfrac{x^4}{4}+C$

= 2/3 x^3/2 +x⁴/4+C, where C is an integral constant.

Also Read:

Integration: Definition, Formulas, Examples