Question: 3) Find the equation of the tangent line to
f(x)=3x³-4x² at x=-1
4) Find the equation of the tangent line to
f(x)=√x @ x=2
5) Find the equation of the tangent line to
f(x)=1/x @ x=2
5) Find the equation of the tangent line to
f(x)=2x^5 @ x=-2
1. **Problem:** Find the equation of the tangent line to the function $$f(x)=3x^3-4x^2$$ at $$x=-1$$.
2. **Formula:** The equation of the tangent line at $$x=a$$ is given by:
$$y = f'(a)(x - a) + f(a)$$
where $$f'(a)$$ is the derivative of $$f(x)$$ evaluated at $$x=a$$.
3. **Find the derivative:**
$$f(x) = 3x^3 - 4x^2$$
$$f'(x) = \frac{d}{dx}(3x^3) - \frac{d}{dx}(4x^2) = 9x^2 - 8x$$
4. **Evaluate the derivative at $$x=-1$$:**
$$f'(-1) = 9(-1)^2 - 8(-1) = 9 + 8 = 17$$
5. **Evaluate the function at $$x=-1$$:**
$$f(-1) = 3(-1)^3 - 4(-1)^2 = 3(-1) - 4(1) = -3 - 4 = -7$$
6. **Write the tangent line equation:**
$$y = 17(x - (-1)) + (-7) = 17(x + 1) - 7$$
7. **Simplify:**
$$y = 17x + 17 - 7 = 17x + 10$$
**Final answer:**
$$\boxed{y = 17x + 10}$$