Sketching functions

Find the points of intersection of the graph of $f(x) = x^3 - 4x$ with the coordinate axes.

Check that the function $f(x) = x^3 - x$ is odd and explain the advantage of this symmetry when sketching its graph.

Analyse and sketch the biquadratic function $f(x) = x^4 - 2x^2$: symmetry, axis intercepts, intervals of monotonicity, local extrema and a sketch of the graph.

You can do it on paper and send a photo with your answer.

Determine the domain of the function $f(x) = \frac{x + 1}{x - 3}$ and state its vertical asymptote, studying the one-sided limits at the point that does not belong to the domain.

Analyse and sketch the rational function $f(x) = \frac{x - 1}{x + 2}$: domain, intercepts, vertical and horizontal asymptotes with their limits, monotonicity using the derivative and a sketch of the graph.

You can do it on paper and send a photo with your answer.

Carry out a complete analysis and sketch the graph of $f(x) = \frac{x^2}{x - 2}$:

1. a) Domain, axis intercepts and symmetries.

1. b) Vertical, horizontal and oblique asymptotes, with the corresponding limits.

1. c) Intervals of monotonicity and local extrema.

1. d) Position of the curve relative to the oblique asymptote and final sketch.

Analyse and sketch the function $f(x) = x^3 - 3x^2$ by following these steps:

1. a) Axis intercepts.

1. b) Derivative, stationary points and intervals of monotonicity.

1. c) Local maximums and minimums.

1. d) Draw the graph using all the previous information.

You can do it on paper and send a photo with your answer.

The temperature of a glass of hot water is measured in a kitchen at $20\ \text{°C}$ and the data in the figure are obtained. The proposed model is $T(t) = 20 + \frac{80}{t + 1}$, where $t$ is in minutes.

1. a) Check that the model is consistent with the first data points in the figure.

1. b) Study the monotonicity of $T(t)$ for $t \geq 0$ using its derivative.

1. c) Calculate the horizontal asymptote and interpret its physical meaning.

1. d) Sketch the graph of the model for $t \geq 0$.

You can do it on paper and send a photo with your answer.

Analyse and sketch the function $f(x) = x + \frac{1}{x}$: domain, odd symmetry, absence of x-intercepts (explain why), vertical and oblique asymptotes, intervals of monotonicity, local extrema with their values, and sketch.

You can do it on paper and send a photo with your answer.

An engineer is designing the profile of a roller coaster. The first descending section follows the equation $y = -x^2 + 4$ for $x \le 1$. To smooth the transition to the climb, they decide to join, at $x = 1$, a second section shaped like the parabola $y = a(x - 2)^2 + b$ for $x > 1$.

For the join to be smooth, the whole function must be continuous and differentiable at $x = 1$ (no jumps or corners). Moreover, the curvature must change at that point: the first section is concave and the second must be convex, so that $x = 1$ is an inflection point.

1. a) Find the values of $a$ and $b$ by imposing continuity and differentiability at $x = 1$.

1. b) Check that with those values the change of curvature at $x = 1$ holds, by studying the sign of the second derivative of each section.

A fundamental geometric property of convex functions ($\cup$) is that the line segment joining any two points on its graph always lies above the curve.

Verify this property analytically for the function $f(x) = x^2$. To do so, find the equation of the secant line passing through the points on the graph with x-coordinates $x = -1$ and $x = 3$. Then, calculate the value of the line at $x = 1$ and check that it is strictly greater than $f(1)$.

Describe and draw in your notebook a sketch of a continuous function $f(x)$ defined on the domain $[-5, 5]$ that simultaneously satisfies all of the following conditions:

• It is an odd function (symmetric with respect to the origin).

• It has a relative maximum at $x = 2$ and a relative minimum at $x = -2$.

• It has exactly three inflection points located at $x = -3$, $x = 0$, and $x = 3$.

• It is convex ($\cup$) on the intervals $(-3, 0) \cup (3, 5]$.

• It is concave ($\cap$) on the intervals $[-5, -3) \cup (0, 3)$.

You can do it on paper and submit a photo of your answer.