https://file.aiquickdraw.com/mathbot/file/17212463260292nnfjtej.jpg

To find the instantaneous velocity of the object at \( t = 1 \), we need to use the definition of the derivative, which gives us the instantaneous rate of change of the position function \( s(t) \). The formula for the instantaneous velocity at a given time \( t = a \) is:

\[ v(a) = \lim_{h \to 0} \frac{s(a+h) - s(a)}{h} \]

This corresponds to option C in the multiple-choice list.

To find the instantaneous velocity at \( t = 1 \), we need to differentiate the given position function \( s(t) = 4t^2 + 3t + 4 \) and then evaluate the derivative at \( t = 1 \).

First, let's differentiate \( s(t) \):

\[ s(t) = 4t^2 + 3t + 4 \]

Using the power rule for differentiation, we get:

\[ s'(t) = \frac{d}{dt}(4t^2) + \frac{d}{dt}(3t) + \frac{d}{dt}(4) \]
\[ s'(t) = 8t + 3 \]

Now, we substitute \( t = 1 \) into the derivative:

\[ s'(1) = 8(1) + 3 \]
\[ s'(1) = 8 + 3 \]
\[ s'(1) = 11 \]

Therefore, the instantaneous velocity at \( t = 1 \) is \( 11 \).

Would you like any further details or have any questions?

Here are 5 numbered questions you might ask next:
1. How do you find the derivative of a function?
2. What are the steps to solve limits?
3. Can you explain the power rule for differentiation?
4. What is the difference between average velocity and instantaneous velocity?
5. How do you interpret the derivative in a real-world context?

**Tip:** Always remember to check the differentiation rules when solving problems involving rates of change and instantaneous velocities.

Learn how to find the instantaneous velocity of an object at t = 1 using calculus. This problem involves the differentiation of a position function and application of the derivative to find the rate of change. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation