Find the equation of the normal line to the curve $f(x)=2 x^3-3 x+1$ at the point $(1,0)$.

Based on your answers to the previous questions, how are the different possible graphs of f related to one another?

The tangent to the curve $y=\frac{2 x}{x-1}, x \neq 1$ at the point $(2,4)$ and the normal to the same curve at the point $(3,3)$ intersect at the point $P$. Find the coordinates of $P$.

Find:

a. any points of inflexion

b. the intervals where the function is concave up

c. the intervals where the function is concave down

Justify your answer.

$$y=2 x^4+x^3+1$$

Plot the graph of $g(x)=x+1$ on your GDC. Evaluate $g(x)$ at each of the points where you evaluated $f(x)$ in the previous question.

Consider the graph of a quadratic function whose leading coefficient is negative, that is, a < 0.

For a < 0, the vertex is a maximum point.

What sign do the gradients left of the maximum point on the parabola have?

Based on your answers from the previous questions, does Achilles overtake the tortoise?

Let $f(x)=5 a x^3-2 b x^2+4 c x$.

a. Find $f^{\prime}(x)$.

b. Given that $f^{\prime}(x) \geq 0$ and $a>0$, show that $b^2 \leq 15 a c$.

What are the intervals for which f is i. concave up and ii. concave down? What does this tell you about the values of $f^{\prime \prime}$ in these intervals?

How is the power rule useful in differentiating a polynomial function?

Compare the expression you obtained for $\frac{\mathrm{d} y}{\mathrm{~d} u} \cdot \frac{\mathrm{d} u}{\mathrm{~d} x}$ to the expression you found for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in the previous question.

Based on your answers from the previous questions, can you find the sum to infinity of this series?

What is optimization in calculus?

Based on your answers to the previous questions, state the signs of $f^{\prime \prime}$ to the left and to the right of this point on f.