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Let's go through each part of the problem step by step. The function given is:

\[
f(t) = 5t^2 + 7t
\]

### a) Find \( f(t + h) \)

We substitute \( t + h \) into the given function for \( t \):

\[
f(t + h) = 5(t + h)^2 + 7(t + h)
\]

Expanding both terms:

\[
= 5(t^2 + 2th + h^2) + 7(t + h)
\]

\[
= 5t^2 + 10th + 5h^2 + 7t + 7h
\]

Thus, 

\[
f(t + h) = 5t^2 + 10th + 5h^2 + 7t + 7h
\]

### b) Find \( f(t + h) - f(t) \)

Now we subtract \( f(t) \) from \( f(t + h) \):

\[
f(t + h) - f(t) = (5t^2 + 10th + 5h^2 + 7t + 7h) - (5t^2 + 7t)
\]

Simplifying:

\[
= 10th + 5h^2 + 7h
\]

### c) Find \( \frac{f(t + h) - f(t)}{h} \)

Now divide the result from part (b) by \( h \):

\[
\frac{f(t + h) - f(t)}{h} = \frac{10th + 5h^2 + 7h}{h}
\]

Simplifying:

\[
= 10t + 5h + 7
\]

### d) Find \( f'(t) \)

The derivative \( f'(t) \) is calculated by differentiating \( f(t) = 5t^2 + 7t \):

\[
f'(t) = \frac{d}{dt}(5t^2 + 7t) = 10t + 7
\]

---

Would you like further details or clarification on any of the steps? Here are some related questions for practice:

1. What would \( f(t + 2h) \) look like?
2. What is the second derivative of \( f(t) \)?
3. How does the result of \( \frac{f(t + h) - f(t)}{h} \) relate to the definition of the derivative?
4. Can you express \( f(t + h) \) using a binomial expansion?
5. How would the derivative change if the function was \( f(t) = 5t^3 + 7t \)?

**Tip**: The process in part (c) is essentially finding the difference quotient, which is a core concept in defining the derivative of a function.

Let f(t) = 5t^2 + 7t. Find f(t + h), f(t + h) - f(t), (f(t+h) - f(t)) / h, and f'(t).

Learn how to find f(t + h), the difference quotient, and the derivative for a quadratic function f(t) = 5t^2 + 7t. This step-by-step guide covers algebraic expansion, simplification, and calculus concepts such as derivatives. Suitable for high school or early college students.

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