Math Problem Statement

Given the function f(x) = 6x^2 + 5x, calculate the following values: f(a), f(a + h), and (f(a + h) - f(a)) / h.

Solution

Let's break this down and solve each part step by step. The function given is:

f(x)=6x2+5xf(x) = 6x^2 + 5x

1. f(a)f(a)

We need to substitute aa into the function f(x)f(x):

f(a)=6a2+5af(a) = 6a^2 + 5a

2. f(a+h)f(a + h)

Substitute a+ha + h into the function f(x)f(x):

f(a+h)=6(a+h)2+5(a+h)f(a + h) = 6(a + h)^2 + 5(a + h)

Now expand each term:

f(a+h)=6(a2+2ah+h2)+5(a+h)f(a + h) = 6(a^2 + 2ah + h^2) + 5(a + h) f(a+h)=6a2+12ah+6h2+5a+5hf(a + h) = 6a^2 + 12ah + 6h^2 + 5a + 5h

3. f(a+h)f(a)h\frac{f(a + h) - f(a)}{h}

We now calculate f(a+h)f(a)h\frac{f(a + h) - f(a)}{h}:

f(a+h)f(a)h=(6a2+12ah+6h2+5a+5h)(6a2+5a)h\frac{f(a + h) - f(a)}{h} = \frac{(6a^2 + 12ah + 6h^2 + 5a + 5h) - (6a^2 + 5a)}{h}

Simplify the expression inside the numerator:

=6a2+12ah+6h2+5a+5h6a25ah= \frac{6a^2 + 12ah + 6h^2 + 5a + 5h - 6a^2 - 5a}{h} =12ah+6h2+5hh= \frac{12ah + 6h^2 + 5h}{h}

Now factor out hh in the numerator:

=h(12a+6h+5)h= \frac{h(12a + 6h + 5)}{h}

Since h0h \neq 0, cancel out hh:

=12a+6h+5= 12a + 6h + 5

Final answers:

  1. f(a)=6a2+5af(a) = 6a^2 + 5a
  2. f(a+h)=6a2+12ah+6h2+5a+5hf(a + h) = 6a^2 + 12ah + 6h^2 + 5a + 5h
  3. f(a+h)f(a)h=12a+6h+5\frac{f(a + h) - f(a)}{h} = 12a + 6h + 5

Let me know if you'd like more details or have any other questions!

Here are some related questions for practice:

  1. What is f(x)f'(x) (the derivative of f(x)=6x2+5xf(x) = 6x^2 + 5x)?
  2. What happens to f(a+h)f(a)h\frac{f(a+h)-f(a)}{h} as h0h \to 0?
  3. How would you interpret the expression f(a+h)f(a)f(a+h)-f(a) geometrically?
  4. How would f(x)=6x2+5xf(x) = 6x^2 + 5x change if the coefficient of x2x^2 were different?
  5. Can you compute f(x)f'(x) for the general function f(x)=ax2+bxf(x) = ax^2 + bx?

Tip: Always expand and simplify carefully to avoid mistakes when calculating limits or differences in functions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Limits
Difference Quotient

Formulas

f(x) = 6x^2 + 5x
Difference Quotient = (f(a + h) - f(a)) / h

Theorems

Basic function evaluation
Expansion of binomials

Suitable Grade Level

Grades 10-12