Berechnung von Wendepunkten und Wendetangenten – Erweiterte Grafikversion

Übersicht

Zu jeder Funktion werden dargestellt:

• f(x)
• f’(x)
• f’’(x)
• f’’’(x)
• Wendepunkte
• Wendetangenten

---

Aufgabe 1a

f(x) = 4x^4 - 6x^2

f  <- function(x) 4*x^4 - 6*x^2
f1 <- function(x) 16*x^3 - 12*x
f2 <- function(x) 48*x^2 - 12
f3 <- function(x) 96*x

x <- seq(-2, 2, length.out = 1000)
df <- data.frame(x = x)

xw <- c(-0.5, 0.5)
yw <- f(xw)

tan1 <- function(x) f1(xw[1])*(x - xw[1]) + f(xw[1])
tan2 <- function(x) f1(xw[2])*(x - xw[2]) + f(xw[2])

p1 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f(x)), color="blue") +
  geom_point(data=data.frame(x=xw,y=yw), aes(x=x,y=y), color="red", size=3) +
  geom_line(aes(y = tan1(x)), linetype="dashed") +
  geom_line(aes(y = tan2(x)), linetype="dashed") +
  ggtitle("f(x)")

p2 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f1(x)), color="darkgreen") +
  ggtitle("f'(x)")

p3 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f2(x)), color="purple") +
  geom_hline(yintercept = 0, linetype="dotted") +
  ggtitle("f''(x)")

p4 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f3(x)), color="orange") +
  ggtitle("f'''(x)")

grid.arrange(p1, p2, p3, p4, ncol=1)

---

Aufgabe 1d

f(x) = sin(x)

f  <- function(x) sin(x)
f1 <- function(x) cos(x)
f2 <- function(x) -sin(x)
f3 <- function(x) -cos(x)

x <- seq(-pi/2, 3*pi/2, length.out = 1000)
df <- data.frame(x = x)

xw <- c(0, pi)
yw <- f(xw)

tan1 <- function(x) f1(xw[1])*(x - xw[1]) + f(xw[1])
tan2 <- function(x) f1(xw[2])*(x - xw[2]) + f(xw[2])

p1 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f(x)), color="blue") +
  geom_point(data=data.frame(x=xw,y=yw), aes(x=x,y=y), color="red", size=3) +
  geom_line(aes(y = tan1(x)), linetype="dashed") +
  geom_line(aes(y = tan2(x)), linetype="dashed") +
  ggtitle("sin(x)")

p2 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f1(x)), color="darkgreen") +
  ggtitle("cos(x)")

p3 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f2(x)), color="purple") +
  geom_hline(yintercept = 0, linetype="dotted") +
  ggtitle("-sin(x)")

p4 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f3(x)), color="orange") +
  ggtitle("-cos(x)")

grid.arrange(p1, p2, p3, p4, ncol=1)

---

Aufgabe 2a

f(x) = 3x^2 - x^3

f  <- function(x) 3*x^2 - x^3
f1 <- function(x) 6*x - 3*x^2
f2 <- function(x) 6 - 6*x
f3 <- function(x) -6

x <- seq(-1, 3, length.out = 1000)
df <- data.frame(x = x)

xw <- 1
yw <- f(xw)

tan <- function(x) f1(xw)*(x - xw) + f(xw)

p1 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f(x)), color="blue") +
  geom_point(data=data.frame(x=xw,y=yw), aes(x=x,y=y), color="red", size=3) +
  geom_line(aes(y = tan(x)), linetype="dashed") +
  ggtitle("f(x)")

p2 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f1(x)), color="darkgreen") +
  ggtitle("f'(x)")

p3 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f2(x)), color="purple") +
  geom_hline(yintercept = 0, linetype="dotted") +
  ggtitle("f''(x)")

p4 <- ggplot(df, aes(x)) +
  geom_hline(yintercept = -6, color="orange") +
  ggtitle("f'''(x)")

grid.arrange(p1, p2, p3, p4, ncol=1)

---

Aufgabe 2b

f(x) = 2x^2 - 1/3 x^4 + 3

f  <- function(x) 2*x^2 - (1/3)*x^4 + 3
f1 <- function(x) 4*x - (4/3)*x^3
f2 <- function(x) 4 - 4*x^2
f3 <- function(x) -8*x

x <- seq(-2, 2, length.out = 1000)
df <- data.frame(x = x)

xw <- c(-1, 1)
yw <- f(xw)

tan1 <- function(x) f1(xw[1])*(x - xw[1]) + f(xw[1])
tan2 <- function(x) f1(xw[2])*(x - xw[2]) + f(xw[2])

p1 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f(x)), color="blue") +
  geom_point(data=data.frame(x=xw,y=yw), aes(x=x,y=y), color="red", size=3) +
  geom_line(aes(y = tan1(x)), linetype="dashed") +
  geom_line(aes(y = tan2(x)), linetype="dashed") +
  ggtitle("f(x)")

p2 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f1(x)), color="darkgreen") +
  ggtitle("f'(x)")

p3 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f2(x)), color="purple") +
  geom_hline(yintercept = 0, linetype="dotted") +
  ggtitle("f''(x)")

p4 <- ggplot(df, aes(x)) +
  geom_line(aes(y = f3(x)), color="orange") +
  ggtitle("f'''(x)")

grid.arrange(p1, p2, p3, p4, ncol=1)