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

> with(Student[Calculus1]); -1
 

> f := proc (x) options operator, arrow, function_assign; `+`(`*`(`^`(`*`(`^`(x, 2)), `/`(1, 3))), sin(x)) end proc;
 

Typesetting:-mprintslash([f := proc (x) options operator, arrow, function_assign; `+`(`*`(`^`(`*`(`^`(x, 2)), `/`(1, 3))), sin(x)) end proc], [proc (x) options operator, arrow, function_assign; `+`(`*... (7.1.1)
 

> plot(f(x), x = -2 .. 8);
 

Plot_2d
 

>
 

>
 

> f(0);
 

0 (7.1.2)
 

> fsolve(f(x) = 0, x = -2 .. 8);
 

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)
 

>
 

>
 

> diff(f(x), x);
 

`+`(`/`(`*`(`/`(2, 3), `*`(x)), `*`(`^`(`*`(`^`(x, 2)), `/`(2, 3)))), cos(x)) (7.1.7)
 

> fp := proc (x) options operator, arrow, function_assign; `+`(`/`(`*`(`/`(2, 3), `*`(x)), `*`(`^`(`*`(`^`(x, 2)), `/`(2, 3)))), cos(x)) end proc; -1; 'fp(x)'
 

fp(x) (7.1.8)
 

> TestList1 := [-1, -.5, -.25, 1, 5];
 

Typesetting:-mprintslash([TestList1 := [-1, -.5, -.25, 1, 5]], [[-1, -.5, -.25, 1, 5]]) (7.1.9)
 

> map(fp, TestList1);
 

[`+`(`-`(`/`(2, 3)), cos(1)), 0.376351952e-1, -0.893549463e-1, `+`(`/`(2, 3), cos(1)), `+`(`*`(`/`(2, 15), `*`(`^`(25, `/`(1, 3)))), cos(5))] (7.1.10)
 

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

[-.12637, 0.37635e-1, -0.89355e-1, 1.2070, .67352] (7.1.11)
 

>
 

>
 

>
 

>
 

> diff(fp(x), x);
 

`+`(`-`(`/`(`*`(`/`(8, 9), `*`(`^`(x, 2))), `*`(`^`(`*`(`^`(x, 2)), `/`(5, 3))))), `/`(`*`(`/`(2, 3)), `*`(`^`(`*`(`^`(x, 2)), `/`(2, 3)))), `-`(sin(x))) (7.1.12)
 

> fpp := proc (x) options operator, arrow, function_assign; `+`(`-`(`/`(`*`(`/`(8, 9), `*`(`^`(x, 2))), `*`(`^`(`*`(`^`(x, 2)), `/`(5, 3))))), `/`(`*`(`/`(2, 3)), `*`(`^`(`*`(`^`(x, 2)), `/`(2, 3)))), `...
 

fpp(x) (7.1.13)
 

> 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] (7.1.14)
 

>
 

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

Typesetting:-mprintslash([TestList2 := [-.75, -.25, .25, 2, 3.25, 5, 6.27]], [[-.75, -.25, .25, 2, 3.25, 5, 6.27]]) (7.1.15)
 

> 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... (7.1.16)
 

> evalf[5]([.3555224887, -1.163619197, -1.658427115, `+`(`-`(`*`(`/`(1, 18), `*`(`^`(4, `/`(1, 3))))), `-`(sin(2))), 0.620340144e-1, `+`(`-`(`*`(`/`(2, 225), `*`(`^`(25, `/`(1, 3))))), `-`(sin(5))), -0....
 

[.35552, -1.1636, -1.6584, -.99749, 0.62034e-1, .93293, -0.60355e-2] (7.1.17)
 

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