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Ejercicios — Tema 2: Cálculo diferencial e integral en una variable

1. Calcular los siguientes límites:

   a. $\underset{x\to \mathrm{\infty }}{lim}\frac{-2x^3-5x+6}{x^2+2x-8}$

   b. $\underset{x\to -\mathrm{\infty }}{lim}\frac{-7x^3+3x-6}{x^2+5x-4}$

   c. $\underset{x\to \mathrm{\infty }}{lim}\frac{x^4-3x+2}{x^4+2x-8}$

   d. $\underset{x\to \mathrm{\infty }}{lim}\frac{2x+6}{x^2+2x-8}$

   e. $\underset{x\to -\mathrm{\infty }}{lim}\frac{x^3+x+1}{x^4+5x-4}$

   f. $\underset{x\to \mathrm{\infty }}{lim}\frac{x^4-3x+2}{x^4+2x^2+x-8}$

   g. $\underset{x\to \mathrm{\infty }}{lim}\frac{-x^5+x+7}{x^4+2x^2-8x+1}$

   h. $\underset{x\to -\mathrm{\infty }}{lim}\frac{x^3-6}{x^4+x^2}$

   i. $\underset{x\to \mathrm{\infty }}{lim}\frac{x^2-3x+2}{x^3+2x^2-8}$

2. Calcula para la siguiente función:

   $$f(x)=\{\begin{array}{ll}\frac{x^2}{x^2-1}& \text{si }x<1\\ \frac{3x}{x-3}& \text{si }x\ge 1\end{array}$$

   $\underset{x\to -\mathrm{\infty }}{lim}f(x)$, $\underset{x\to \mathrm{\infty }}{lim}f(x)$, $\underset{x\to -1}{lim}f(x)$, $\underset{x\to 3}{lim}f(x)$, $\underset{x\to 0}{lim}f(x)$, $\underset{x\to 1}{lim}f(x)$

3. Calcula para la siguiente función:

   $$f(x)=\{\begin{array}{ll}\frac{x^2}{x^2-4}& \text{si }x<0\\ 3x-9& \text{si }0\le x<2\\ 3& \text{si }x>2\end{array}$$

   $\underset{x\to -\mathrm{\infty }}{lim}f(x)$, $\underset{x\to \mathrm{\infty }}{lim}f(x)$, $\underset{x\to 0}{lim}f(x)$, $\underset{x\to 2}{lim}f(x)$, $\underset{x\to 1}{lim}f(x)$

4. Estudiar la continuidad de las siguientes funciones:

   a.

   $$f(x)=\{\begin{array}{ll}-x^2+4& \text{si }x<1\\ 2x-1& \text{si }x\ge 1\end{array}$$

   b.

   $$f(x)=\{\begin{array}{ll}-x^2+4& \text{si }x<1\\ 2x+1& \text{si }x\ge 1\end{array}$$

   c.

   $$f(x)=\{\begin{array}{ll}-x^2-3x& \text{si }x<0\\ -2x+1& \text{si }x\ge 0\end{array}$$

   d.

   $$f(x)=\{\begin{array}{ll}x+1& \text{si }x<-1\\ x^2-x-2& \text{si }x\ge -1\end{array}$$

   e.

   $$f(x)=\{\begin{array}{ll}{x}^2-2& \text{si }x<0\\ \frac{1}{x-1}& \text{si }0\le x<2\\ 2x+2& \text{si }x\ge 2\end{array}$$

   f.

   $$f(x)=\{\begin{array}{ll}{x}^2+2& \text{si }x<1\\ 3& \text{si }x=1\\ 2x+2& \text{si }x>1\end{array}$$

5. Calcular la derivada de las siguientes funciones.

   (1) $y=x^4+3x^2-6$

   (2) $y=6x^3-x^2$

   (3) $y=\frac{x^5}{a+b}-\frac{x^2}{a-b}$

   (4) $y=\frac{x^3-x^2+1}{5}$

   (5) $y=2a{x}^3-\frac{x^2}{b}+c$

   (6) $y=6x^{\frac{7}{2}}+4x^{\frac{5}{2}}+2x$

   (7) $y=\sqrt{3x}+\sqrt[3]{x}+\frac{1}{x}$

   (8) $y=\frac{(x+1{)}^3}{x^{\frac{3}{2}}}$

   (9) $y=\sqrt[3]{x^2}-2\sqrt{x}+5$

   (10) $y=\frac{a{x}^2}{\sqrt[3]{x}}+\frac{b}{x\sqrt{x}}-\frac{\sqrt[3]{x}}{\sqrt{x}}$

   (11) $y=(1+4x^3)(1+2x^2)$

   (12) $y=x(2x-1)(3x+2)$

   (13) $y=(2x-1)(x^2-6x+3)$

   (14) $y=\frac{2x^4}{b^2-x^2}$

   (15) $y=\frac{a-x}{a+x}$

   (16) $f(t)=\frac{t^3}{1+t^2}$

   (17) $f(s)=\frac{(s+4{)}^2}{s+3}$

   (18) $y=\frac{x^3+1}{x^2-x-2}$

   (19) $y=(2x^2-3{)}^2$

   (20) $y=(x^2+a^2{)}^5$

   (21) $y=\sqrt{x^2+a^2}$

   (22) $y=(a+x)\sqrt{a-x}$

   (23) $y=\sqrt{\frac{1+x}{1-x}}$

   (24) $y=\frac{2x^2-1}{x\sqrt{1+x^2}}$

   (25) $y=\sqrt[3]{x^2+x+1}$

   (26) $y=(1+\sqrt[3]{x}{)}^3$

   (27) $y={\sin }^{2}x$

   (28) $y=2\sin x+\cos 3x$

   (29) $y=\tan (ax+b)$

   (30) $y=\frac{\sin x}{1+\cos x}$

   (31) $y=\sin 2x\cos 3x$

   (32) $y={\cot }^{2}5x$

   (33) $f(t)=t\sin t+\cos t$

   (34) $f(t)={\sin }^{3}t\cos t$

   (35) $y=a\sqrt{\cos 2x}$

   (36) $y=\frac{1}{2}{\tan }^{2}x$

   (37) $y=\ln \cos x$

   (38) $y=\ln \tan x$

   (39) $y=\ln {\sin }^{2}x$

   (40) $y=\frac{\tan x-1}{\sec x}$

   (41) $y=\ln \sqrt{\frac{1+\sin x}{1-\sin x}}$

   (42) $f(x)=\sin (\ln x)$

   (43) $f(x)=\tan (\ln x)$

   (44) $f(x)=\sin (\cos x)$

   (45) $y=\ln \frac{1+x}{1-x}$

   (46) $y={\log }_3(x^2-\sin x)$

   (47) $y=\ln \frac{1+x^2}{1-x^2}$

   (48) $y=\ln (x^2+x)$

   (49) $y=\ln (x^3-2x+5)$

   (50) $y=x\ln x$

   (51) $y={\ln }^{3}x$

   (52) $y=\ln (x+\sqrt{1+x^2})$

   (53) $y=\ln (\ln x)$

   (54) $y=e^{4x+5}$

   (55) $y=a^{x^2}$

   (56) $y=7^{x^2+2x}$

   (57) $y=e^x(1-x^2)$

   (58) $y=\frac{e^x-1}{e^x+1}$

   (59) $y=e^{\sin x}$

   (60) $y=a^{\tan nx}$

   (61) $y=e^{\cos x}\sin x$

   (62) $y=e^x\ln (\sin x)$

   (63) $y=3x^3-5x^2+6x-2$

   (64) $y=(3x^2-1)(4x^3+2)$

   (65) $y=3x^2+4e^x-5\ln x-6\cos x+10\sin x$

   (66) $y=4x^2+(3x^2-2x)(4x^3+2)$

   (67) $y=\frac{x^2-2}{x^2-9}$

   (68) $y=\frac{3x^2}{x^2-4}$

   (69) $y=\frac{2x^2}{x-4}$

   (70) $y=\frac{3x^2}{2x^2+4}$

   (71) $y=\frac{3x^3}{x^2-1}$

   (72) $y=\frac{3x^2}{x^2-3x}$

   (73) $y=\frac{2x^2-1}{2x^2}$

   (74) $y=\frac{2x^2-1}{x}$

   (75) $y=(3x^2-1{)}^3$

   (76) $y=\sqrt{2x^3+3}$

   (77) $y=x^2{e}^{3x}$

   (78) $y=\ln (x^2+1)$

   (79) $y=x\sqrt{2x+3}$

   (80) $y=\sin (x^2-3)\ln (x^3+1)$

   (81) $y=\frac{2x^2-1}{2e^{3x}}$

   (82) $y=\sin (x^2+3)\tan (e^x)$

6. Aplicaciones de la derivada.

   (1) Determinar las tangentes de los ángulos que forman con el eje positivo de las $x$ las líneas tangentes a la curva $y=x^3$ cuando $x=1/2$ y $x=-1$, construir la gráfica y representar las líneas tangentes.

   (2) Determinar las tangentes de los ángulos que forman con el eje positivo de las $x$ las líneas tangentes a la curva $y=1/x$ cuando $x=1/2$ y $x=1$, construir la gráfica y representar las líneas tangentes.

   (3) Calcular los extremos relativos de $f(x)=\frac{x^2}{x+2}$

   (4) Calcular los extremos relativos de $f(x)=x^3-15x^2+72x+8$

   (5) Calcular los extremos relativos de $f(x)=\frac{x^2-1}{x^2+2}$

   (6) Calcular los extremos relativos de $f(x)=\frac{(x-1)(x-2)}{x^2}$

7. Calcular las siguientes integrales.

   (1) $\int x^{5}dx$

   (2) $\int (x+\sqrt{x})dx$

   (3) $\int (\frac{3}{\sqrt{x}}-\frac{x\sqrt{x}}{4})dx$

   (4) $\int \frac{x^2}{\sqrt{x}}dx$

   (5) $\int (\frac{1}{x^2}+\frac{4}{x\sqrt{x}}+2)dx$

   (6) $\int \frac{1}{\sqrt[4]{x}}dx$

   (7) $\int e^{5x}dx$

   (8) $\int \cos 5xdx$

   (9) $\int \sin axdx$

   (10) $\int \frac{\ln x}{x}dx$

   (11) $\int \frac{1}{{\sin }^{2}3x}dx$

   (12) $\int \frac{1}{{\cos }^{2}7x}dx$

   (13) $\int \frac{1}{3x-7}dx$

   (14) $\int \frac{1}{1-x}dx$

   (15) $\int \frac{1}{5-2x}dx$

   (16) $\int \tan 2xdx$

   (17) $\int {\sin }^{2}x\cos xdx$

   (18) $\int {\cos }^{3}x\sin xdx$

   (19) $\int x\sqrt{x^2+1}dx$

   (20) $\int \frac{x}{\sqrt{2x^2+3}}dx$

   (21) $\int \frac{\cos x}{{\sin }^{2}x}dx$

   (22) $\int \frac{\sin x}{{\cos }^{2}x}dx$

   (23) $\int \frac{\tan x}{{\cos }^{2}x}dx$

   (24) $\int \frac{\cot x}{{\sin }^{2}x}dx$

   (25) $\int \frac{\ln (x+1)}{x+1}dx$

   (26) $\int \frac{\cos x}{\sqrt{2\sin x+1}}dx$

   (27) $\int \frac{\sin 2x}{(1+\cos 2x{)}^2}dx$

   (28) $\int \frac{\sin 2x}{\sqrt{1+{\sin }^{2}x}}dx$

   (29) $\int \frac{\sqrt{\tan x+1}}{{\cos }^{2}x}dx$

   (30) $\int \frac{{\ln }^{2}x}{x}dx$

   (31) $\int \frac{\arcsin x}{\sqrt{1-x^2}}dx$

   (32) $\int \frac{x}{x^2+1}dx$

   (33) $\int \frac{x+1}{x^2+2x+3}dx$

   (34) $\int e^{2x}dx$

   (35) $\int e^{x}dx$

   (36) $\int e^{\sin x}\cos xdx$

   (37) $\int 3^x{e}^{x}dx$

   (38) $\int e^{-3x}dx$

   (39) $\int e^{x^2+4x+3}(x+2)dx$

   (40) $\int \frac{1}{1+2x^2}dx$

   (41) $\int \frac{1}{\sqrt{1-3x^2}}dx$

   (42) $\int \frac{1}{\sqrt{9-x^2}}dx$

   (43) $\int \frac{1}{4+x^2}dx$

   (44) $\int x{e}^{x^2}dx$

   (45) $\int \ln xdx$

   (46) $\int x^2{e}^{3x}dx$

   (47) $\int x^2{e}^{-x}dx$

   (48) $\int x\sin xdx$

   (49) $\int x^2\cos (2x)dx$

   (50) $\int e^x\sin xdx$

   (51) $\int x^3{e}^{x^2}dx$

   (52) $\int \sqrt[3]{x^2}dx$

   (53) $\int \frac{1}{x^2}dx$

   (54) $\int (3x^2+5x+3+\frac{2x+7}{x^2})dx$

   (55) $\int \frac{1+\ln x}{x}dx$

   (56) $\int \frac{e^x}{1+e^x}dx$

   (57) $\int e^{4x}dx$

   (58) $\int \frac{\sin x}{1+\cos x}dx$

   (59) $\int \frac{\arcsin x}{\sqrt{1-x^2}}dx$

   (60) $\int \frac{x^3}{1+x^4}dx$

   (61) $\int x{e}^{x}dx$

   (62) $\int x^4\ln xdx$

   (63) $\int (x^2-1)\cos (2x)dx$

   (64) $\int e^x\sin (3x)dx$

   (65) $\int \ln xdx$

   (66) $\int x^3\ln (3x)dx$

   (67) $\int \frac{2}{x^2-1}dx$

   (68) $\int \frac{5x-4}{x^2-x-2}dx$

   (69) $\int \frac{5x^2-5x+2}{x^3-x^2-x+1}dx$

   (70) $\int \frac{2x^3+3x+3}{x^3+2x^2+x+2}dx$

   (71) $\int \frac{x+2}{x^3-x^2+2x-2}dx$

   (72) $\int \frac{x+2}{x^2+1}dx$

   (73) $\int (x^3-2)e^{x}dx$

   (74) $\int \frac{dx}{x\ln x}$

   (75) $\int \frac{e^x}{1+e^{2x}}dx$

   (76) $\int \frac{2x^2+3x-1}{x^3+5x^2+7x+3}dx$

   (77) $\int {\cos }^3(x)dx$

   (78) $\int (x^3+2x+1)\ln xdx$

   (79) $\int \frac{1-(\ln x{)}^2}{x}dx$

   (80) $\int \frac{x^2}{1+x^3}dx$

   (81) $\int \frac{x^2-3x+4}{x^3-3x^2+3x-1}dx$

8. Calcular las siguientes integrales definidas.

   (1) ${\int }_1^2(x^2-2x+3)dx$

   (2) ${\int }_0^8(\sqrt{2x}+\sqrt[3]{x})dx$

   (3) ${\int }_0^1\frac{x}{x^2+3x+2}dx$

   (4) ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\tan xdx$

   (5) ${\int }_1^4\frac{1+\sqrt{x}}{x^2}dx$

   (6) ${\int }_e^{e^2}\frac{dx}{x\ln x}$

9. Aplicaciones de la integral: cálculo de áreas.

   (1) Calcular el área de la figura limitada por la parábola $y=4x-x^2$ y el eje de abcisas.

   (2) Calcular el área de la figura comprendida entre las parábolas $y=x^2$, $y=\frac{x^2}{2}$ y la recta $y=2x$.

   (3) Calcular el área del segmento de parábola $y=x^2$, que corta a la recta $y=4-\frac{2}{3}{x}^2$.