Solved on Jan 29, 2024

Use f(x)=4x2f(x)=4x^2 to complete the table. Enter the answer as a whole number, fraction, or decimal rounded to 2 decimals. For fractions, use 12\frac{1}{2} or 12-\frac{1}{2} format.

STEP 1

Assumptions
1. The function given is f(x)=4x2 f(x) = 4x^2 .
2. We need to calculate the value of the expression f(x+h)f(x)h \frac{f(x+h)-f(x)}{h} for various values of h h when x=5 x = 5 .
3. The results should be expressed as whole numbers, fractions, or decimals rounded to two decimal places.

STEP 2

First, let's write down the general form of the expression we need to calculate.
f(x+h)f(x)h=4(x+h)24x2h \frac{f(x+h)-f(x)}{h} = \frac{4(x+h)^2 - 4x^2}{h}

STEP 3

Now, let's expand the expression (x+h)2 (x+h)^2 .
(x+h)2=x2+2xh+h2 (x+h)^2 = x^2 + 2xh + h^2

STEP 4

Substitute the expanded form back into the expression.
f(x+h)f(x)h=4(x2+2xh+h2)4x2h \frac{f(x+h)-f(x)}{h} = \frac{4(x^2 + 2xh + h^2) - 4x^2}{h}

STEP 5

Distribute the 4 across the terms inside the parentheses.
f(x+h)f(x)h=4x2+8xh+4h24x2h \frac{f(x+h)-f(x)}{h} = \frac{4x^2 + 8xh + 4h^2 - 4x^2}{h}

STEP 6

Notice that 4x2 4x^2 and 4x2 -4x^2 cancel each other out.
f(x+h)f(x)h=8xh+4h2h \frac{f(x+h)-f(x)}{h} = \frac{8xh + 4h^2}{h}

STEP 7

Factor out h h from the numerator.
f(x+h)f(x)h=h(8x+4h)h \frac{f(x+h)-f(x)}{h} = \frac{h(8x + 4h)}{h}

STEP 8

Cancel out h h from the numerator and denominator.
f(x+h)f(x)h=8x+4h \frac{f(x+h)-f(x)}{h} = 8x + 4h

STEP 9

Now we will evaluate this expression for x=5 x = 5 and various values of h h .

STEP 10

First, let's calculate it for h=2 h = 2 .
f(5+2)f(5)2=85+42 \frac{f(5+2)-f(5)}{2} = 8 \cdot 5 + 4 \cdot 2

STEP 11

Perform the multiplication.
f(5+2)f(5)2=40+8 \frac{f(5+2)-f(5)}{2} = 40 + 8

STEP 12

Add the results.
f(5+2)f(5)2=48 \frac{f(5+2)-f(5)}{2} = 48
This matches the value given in the table for h=2 h = 2 .

STEP 13

Next, calculate it for h=1 h = 1 .
f(5+1)f(5)1=85+41 \frac{f(5+1)-f(5)}{1} = 8 \cdot 5 + 4 \cdot 1

STEP 14

Perform the multiplication.
f(5+1)f(5)1=40+4 \frac{f(5+1)-f(5)}{1} = 40 + 4

STEP 15

Add the results.
f(5+1)f(5)1=44 \frac{f(5+1)-f(5)}{1} = 44
This matches the value given in the table for h=1 h = 1 .

STEP 16

Now, calculate it for h=0.1 h = 0.1 .
f(5+0.1)f(5)0.1=85+40.1 \frac{f(5+0.1)-f(5)}{0.1} = 8 \cdot 5 + 4 \cdot 0.1

STEP 17

Perform the multiplication.
f(5+0.1)f(5)0.1=40+0.4 \frac{f(5+0.1)-f(5)}{0.1} = 40 + 0.4

STEP 18

Add the results.
f(5+0.1)f(5)0.1=40.4 \frac{f(5+0.1)-f(5)}{0.1} = 40.4
This matches the value given in the table for h=0.1 h = 0.1 .

STEP 19

Finally, calculate it for h=0.01 h = 0.01 .
f(5+0.01)f(5)0.01=85+40.01 \frac{f(5+0.01)-f(5)}{0.01} = 8 \cdot 5 + 4 \cdot 0.01

STEP 20

Perform the multiplication.
f(5+0.01)f(5)0.01=40+0.04 \frac{f(5+0.01)-f(5)}{0.01} = 40 + 0.04

STEP 21

Add the results.
f(5+0.01)f(5)0.01=40.04 \frac{f(5+0.01)-f(5)}{0.01} = 40.04
This matches the value given in the table for h=0.01 h = 0.01 .

STEP 22

Now we have verified all the given values in the table and found that they are correct.
The completed table with the expression f(x+h)f(x)h \frac{f(x+h)-f(x)}{h} evaluated for x=5 x = 5 and various h h is:
\begin{tabular}{|c|c|c|} \hline x\mathbf{x} & h\boldsymbol{h} & f(x+h)f(x)h\frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & 48 \\ \hline 5 & 1 & 44 \\ \hline 5 & 0.1 & 40.4 \\ \hline 5 & 0.01 & 40.04 \\ \hline \end{tabular}

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord