1. Perform a full curve sketching analysis on f(x) = `+`(`*`(`^`(`*`(`^`(x, 2)), `/`(1, 3))), sin(x)); limited to the interval   

restart;  

with(Student[Calculus1]); -1 

 

`+`(`*`(`^`(`*`(`^`(x, 2)), `/`(1, 3))), sin(x)); f(x);  

plot(f(x), x = -2 .. 8);  

Plot_2d
 

 

 

f(0);  

0; (7.1.1)
 

f(x) = 0;  

`+`(`*`(`^`(`*`(`^`(x, 2)), `/`(1, 3))), sin(x)) = 0; (7.1.2)
 

 

0.; (7.1.3)
 

 

 

 

Asymptotes(f(x));  

[]; (7.1.4)
 

 

 

CriticalPoints(f(x), x = -2 .. 8);  

 

Warning, some critical points are returned as numeric approximations
[-.7441055484, -.3624502385, 0, 2.116736754, 4.289604449]; (7.1.5)
 

 

 

ExtremePoints(f(x), x = -2 .. 8);  

 

Warning, some extreme points are returned as numeric approximations
[-2, -.7441055484, -.3624502385, 0, 2.116736754, 4.289604449, 8]; (7.1.6)
 

 

 

f(x);  

`+`(`*`(`^`(`*`(`^`(x, 2)), `/`(1, 3))), sin(x)); (7.1.7)
 

 

`+`(`/`(`*`(`/`(2, 3), `*`(x)), `*`(`^`(`*`(`^`(x, 2)), `/`(2, 3)))), cos(x)); (7.1.8)
 

 

fp(x); (7.1.9)
 

 

`+`(`-`(`/`(`*`(`/`(8, 9), `*`(`^`(x, 2))), `*`(`^`(`*`(`^`(x, 2)), `/`(5, 3))))), `/`(`*`(`/`(2, 3)), `*`(`^`(`*`(`^`(x, 2)), `/`(2, 3)))), `-`(sin(x))); (7.1.10)
 

 

fpp(x); (7.1.11)
 

TestList1 := [-1, -.5, -.25, 1, 5];  

TestList1 := [-1, -.5, -.25, 1, 5]; (7.1.12)
 

map(fp, TestList1);  

[`+`(`-`(`/`(2, 3)), cos(1)), 0.376351952e-1, -0.893549463e-1, `+`(`/`(2, 3), cos(1)), `+`(`*`(`/`(2, 15), `*`(`^`(25, `/`(1, 3)))), cos(5))]; (7.1.13)
 

 

[-.12637, 0.37635e-1, -0.89355e-1, 1.2070, .67352]; (7.1.14)
 

 

 

 

 

CriticalPoints(fp(x), x = -2 .. 8);  

 

Warning, some critical points are returned as numeric approximations
[-.5358533706, .5358533706, 3.092245148, 3.188953445, 6.263938843, 6.302275808]; (7.1.15)
 

 

 

TestList2 := [-.75, -.25, .25, 2, 3.25, 5, 6.27];  

TestList2 := [-.75, -.25, .25, 2, 3.25, 5, 6.27]; (7.1.16)
 

map(fpp, TestList2);  

[.3555224887, -1.163619197, -1.658427115, `+`(`-`(`*`(`/`(1, 18), `*`(`^`(4, `/`(1, 3))))), `-`(sin(2))), 0.620340144e-1, `+`(`-`(`*`(`/`(2, 225), `*`(`^`(25, `/`(1, 3))))), `-`(sin(5))), -0.603554908... (7.1.17)
 

 

[.35552, -1.1636, -1.6584, -.99749, 0.62034e-1, .93293, -0.60355e-2]; (7.1.18)