Suppose that f(x) = 7x ^ 2 - 5
(A) Find the slope of the line tangent to f(x) at x = 3
(B) Find the instantaneous rate of change of f(x) at x = 3
(C) Find the equation of the line tangent to f(x) at x = 3 y =

Let's solve each part of the problem step-by-step:

### (A) Find the slope of the line tangent to \( f(x) \) at \( x = 3 \)

The slope of the tangent line at a point is given by the derivative of the function at that point. Let's find the derivative of \( f(x) = 7x^2 - 5 \).

The derivative of \( f(x) \) with respect to \( x \) is:

\[
f'(x) = \frac{d}{dx} (7x^2 - 5)
\]

Calculating the derivative:

\[
f'(x) = 14x
\]

Now, substitute \( x = 3 \) to find the slope at this point:

\[
f'(3) = 14(3) = 42
\]

### Answer to (A): The slope of the tangent line at \( x = 3 \) is \( 42 \).

---

### (B) Find the instantaneous rate of change of \( f(x) \) at \( x = 3 \)

The instantaneous rate of change of a function at a point is the same as the slope of the tangent line at that point. From part (A), we have already found that:

\[
\text{Instantaneous rate of change} = f'(3) = 42
\]

### Answer to (B): The instantaneous rate of change of \( f(x) \) at \( x = 3 \) is \( 42 \).

---

### (C) Find the equation of the line tangent to \( f(x) \) at \( x = 3 \)

To find the equation of the tangent line, we need the slope of the tangent (which is \( 42 \)) and the point at which it touches the curve. The point of tangency is given by the coordinates \((3, f(3))\).

First, find \( f(3) \):

\[
f(3) = 7(3)^2 - 5 = 7(9) - 5 = 63 - 5 = 58
\]

So, the point of tangency is \( (3, 58) \).

Now, use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \( m = 42 \), \( x_1 = 3 \), and \( y_1 = 58 \). Substituting these values:

\[
y - 58 = 42(x - 3)
\]

Simplify the equation:

\[
y - 58 = 42x - 126
\]

\[
y = 42x - 126 + 58
\]

\[
y = 42x - 68
\]

### Answer to (C): The equation of the tangent line at \( x = 3 \) is \( y = 42x - 68 \).

---

Would you like more details or have any questions? Here are some related questions:

1. How do you find the second derivative of a function?
2. What does the second derivative tell you about the function?
3. How do you determine if a tangent line is above or below the curve?
4. What is the physical interpretation of the instantaneous rate of change?
5. How do you find the equation of a normal line to a curve at a given point?

**Tip:** When finding the tangent line, always ensure to plug in the exact coordinates and the correct slope for accurate results.

Suppose that f(x) = 7x ^ 2 - 5 (A) Find the slope of the line tangent to f(x) at x = 3 (B) Find the instantaneous rate of change of f(x) at x = 3 (C) Find the equation of the line tangent to f(x) at x = 3 y =

Solve a calculus problem involving the function f(x) = 7x^2 - 5. Learn how to find the slope and equation of the tangent line at x = 3, as well as the instantaneous rate of change. This detailed solution covers derivatives and is suitable for students in Grades 10-12.

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