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Found in: Page 299

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# The velocity graph of an accelerating car is shown.(a) Estimate the average velocity of the car during the first $$12$$ seconds.(b) At what time was the instantaneous velocity equal to the average velocity?

(a) Average velocity is$$45$$.

(b) It will be equal at $$4.5$$seconds.

See the step by step solution

## Step 2: Average velocity.

(a) Now, find the average velocity of the car during the first $$12$$seconds.

The average value of a function is represented by this graph is,

$${{\rm{f}}_{{\rm{axe}}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{b - a}}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{f(}}{{\rm{x}}_{\rm{i}}}{\rm{)\Delta x}}}$$

Here $$\Delta x = \frac{{b - a}}{n}$$ and $$n$$is a number of terms.

From the graph, $$a = 0,b = 12$$

Let us take $$n = 6$$ and

\begin{aligned}{c}{\rm{\Delta x &= }}\frac{{{\rm{b - a}}}}{{\rm{n}}}\\{\rm{ &= }}\frac{{{\rm{12 - 0}}}}{{\rm{6}}}\\{\rm{ = 2}}\end{aligned}

## Step 3: Average value.

Now, from the graph the average value is

\begin{aligned}{c}{{\rm{f}}_{{\rm{axe}}}}{\rm{ &= }}\frac{{\rm{1}}}{{{\rm{b - a}}}}\sum\limits_{{\rm{i &= 1}}}^{\rm{n}} {{\rm{f'(}}{{\rm{x}}_{\rm{i}}}{\rm{)\Delta x}}} \\{\rm{ &= }}\frac{{\rm{1}}}{{{\rm{12 - 0}}}}\left( {{\rm{\Delta x}}\left( {{\rm{f(1) + f}}\left( {\rm{3}} \right){\rm{ + f}}\left( {\rm{5}} \right){\rm{ + f}}\left( {\rm{7}} \right){\rm{ + f}}\left( {\rm{9}} \right){\rm{ + f}}\left( {{\rm{11}}} \right)} \right)} \right)\\{\rm{ &= }}\frac{{\rm{1}}}{{{\rm{12}}}}{\rm{.2}}\left( {{\rm{10 + 30 + 45 + 55 + 65 + 65}}} \right)\\{\rm{ &= }}\frac{{\rm{1}}}{{\rm{6}}}{\rm{.270}}\\{\rm{ &= 45}}\end{aligned}

Therefore, the average velocity is$$45$$.

## Step 4: Instantaneous velocity.

(b) From the graph,

The velocity of a car was $$45km/h$$ at $$4.5$$seconds.

Therefore, instantaneous velocity is equal to average velocity at $$4.5$$ seconds.

$$t = 4.5$$.