Question: Philippine Christian University
Aguinaldo Highway, Sampaloc 1, Dasmariñas City, Cavite 4114
College of Engineering and Technology
SW 1
Name: Score:
Year & Section: Date:
Instructions: Solve the following problems as instructed. Final answer shall be in its simplest
form and enclosed with a box. Show your solutions. No Solutions wrong.
1. If $f(x) = 2(x + 4)^3 - 5(x + 2)^2 - 5$, find:
a. $f(3x+5)$ ______________________________ (3pts)
b. $f(k^2 - 1)$ ____________________________ (3pts)
c. $f((a + 2b - c))$ _______________________ (3pts)
d. $f((x - 2y)^3)$ _________________________ (3pts)
2. Evaluate the expression $\frac{f(x+h)-f(x)}{h}$ for the following functions;
a. $f(x) = 4x - x^2$ ________________________ (3pts)
b. $f(x) = \sqrt{4x^2}$ ________________________ (3pts)
c. $f(x) = \frac{2}{x}$ ____________________________ (3pts)
3. Determine the limit of the following:
a. $\lim_{x\to 1} \frac{x^4+3x^3-13x^2-27x+36}{x^2+3x-4}$ __________________ (3pts)
b. $\lim_{x\to \frac{\pi}{6}} \frac{\sin 2x}{\sin x \tan x}$ ______________________________ (3pts)
c. $\lim_{x\to \frac{\pi}{2}} \frac{\sin^2 x}{\tan^3 x}$ _______________________________ (3pts)
1. Problem: Given $f(x) = 2(x + 4)^3 - 5(x + 2)^2 - 5$, find $f(3x+5)$.
Step 1: Substitute $3x+5$ into $f(x)$:
$$f(3x+5) = 2((3x+5) + 4)^3 - 5((3x+5) + 2)^2 - 5$$
Step 2: Simplify inside the parentheses:
$$(3x+5) + 4 = 3x + 9$$
$$(3x+5) + 2 = 3x + 7$$
Step 3: Rewrite:
$$f(3x+5) = 2(3x + 9)^3 - 5(3x + 7)^2 - 5$$
Step 4: Expand $(3x + 9)^3$ and $(3x + 7)^2$:
- $(3x + 9)^3 = (3(x + 3))^3 = 27(x + 3)^3$ but we will expand directly:
$$(3x + 9)^3 = (3x)^3 + 3 \times (3x)^2 \times 9 + 3 \times 3x \times 9^2 + 9^3 = 27x^3 + 243x^2 + 729x + 729$$
- $(3x + 7)^2 = 9x^2 + 42x + 49$
Step 5: Substitute expansions:
$$f(3x+5) = 2(27x^3 + 243x^2 + 729x + 729) - 5(9x^2 + 42x + 49) - 5$$
Step 6: Multiply out:
$$= 54x^3 + 486x^2 + 1458x + 1458 - 45x^2 - 210x - 245 - 5$$
Step 7: Combine like terms:
$$54x^3 + (486x^2 - 45x^2) + (1458x - 210x) + (1458 - 245 - 5)$$
$$= 54x^3 + 441x^2 + 1248x + 1208$$
Step 8: Final answer boxed:
$$\boxed{54x^3 + 441x^2 + 1248x + 1208}$$