Ripetizioni di Analisi Analysis 1Unlocked IT Search Log inAccount

Exercises

Graphing functions

Problems to solve on your own. The final solution appears with the button below each problem; each exercise also has its own page, to share with the link.

1. Study of a rational function with an oblique asymptote

Study the function

f (x) = \frac{x ^{2} - 3}{x - 2}

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

2. Study of a rational function with a horizontal asymptote

Study the function

f (x) = \frac{x}{x ^{2} + 1}

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

3. Study of a function with an exponential

Study the function

f (x) = x^{2} e^{x}

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

4. Study of a function with an exponential and an oblique asymptote

Study the function

f (x) = x e^{\frac{1}{x}}

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

5. Study of an even function with a logarithm

Study the function

f (x) = ln (x^{2} + 1)

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

6. Study of a function with a logarithm and a power

Study the function

f (x) = x^{2} ln x

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

7. Study of a function with a root

Study the function

f (x) = x x + 3

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

8. Study of a function with an arctangent

Study the function

f (x) = arctan x - \frac{x}{2}

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

9. Study of a function with an absolute value

Study the function

f (x) = ∣ x - 1∣ e^{x}

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

10. Study of a function with a logarithm and a vertical asymptote

Study the function

f (x) = ln x - x

and sketch its qualitative graph.

Solution available to those who have unlocked the subject.

11. Absolute maximum and minimum on a closed interval

Determine the absolute maximum and minimum of the function

f (x) = x^{3} - 6 x^{2} + 9 x + 1

on the interval $[0, 5]$ .

Solution available to those who have unlocked the subject.

12. Absolute maximum and minimum of a function with an exponential

Determine the absolute maximum and minimum of the function

f (x) = x^{2} e^{- x}

on the interval $[- 1, 4]$ .

Solution available to those who have unlocked the subject.

13. Number of solutions of an equation from the graph

Determine, as $k \in R$ varies, the number of solutions of the equation

x e^{- x} = k .

Solution available to those who have unlocked the subject.

14. Convexity and inflection points of a polynomial

Determine on which intervals the function

f (x) = x^{4} - 4 x^{3} + 10

is convex and on which it is concave, and find its inflection points.

Solution available to those who have unlocked the subject.