Calculus - Basic problems 1

Consider the function $f(x)=x^3+1$.

1. Sketch the graph using a table of values.
2. Determine the $x$-intercept and $y$-intercept of $f$.
3. Where does the graph of $f$ intersect the horizontal line at height $y=9$?
4. Where does the graph of $f$ intersect the graph of the function $x^2+x+1$?
5. Based on your drawing of the graph $f$, estimate the derivative $f^\prime(1)$.
6. Determine $f'(1)$ using the differential quotient.
7. Based on your drawing of the graph $f$, sketch the derivative $f^\prime$. What could be the function equation of $f^\prime$?
8. Calculate the function equation of $f^\prime$ using the rules of differentiation. Use the function equation to determine again $f'(1)$ and compare with (6).
9. Determine the function equation of the tangent to $f$ at $x=1$. Where does the tangent intersect the $x$-axis, where the $y$-axis?
10. Find the stationary points of $f$ and classify them (as local maximum, local minimum, or saddle point) using the higher order derivatives.
11. Estimate the area under the curve of $f$ between $x=0$ and $x=2$ using two bars.
12. Determine the antiderivative of $f$.
13. Determine $\int_0^1 (x^3+1)\, dx$ using the fundamental theorem of calculus.
14. Determine the exact area enclosed by the graph of $f$, the $x$-axis and the vertical lines at $x=0$ and $x=1$.
15. Determine the area enclosed by the graph of $f$, the $x$-axis, and the vertical lines at $x=-2$ and $x=1$.
16. Determine the area enclosed by the graph of $f$ and the graph of $h(x)=4x+1$.

Show