2.        Given f(x) = sqrt(`+`(`-`(`*`(`^`(x, 2))), 36)); determine all values that satisfy the MVT on the interval  [-3, 5]; . 

restart;  

 

f := proc (x) options operator, arrow, function_assign; sqrt(`+`(36, `-`(`*`(`^`(x, 2))))) end proc;  

f := proc (x) options operator, arrow; sqrt(`+`(36, `-`(`*`(`^`(x, 2))))) end proc; (8.2.1)
 

 

 

iscont(f(x), x = -6 .. 6);  

true; (8.2.2)
 

 

limit(`/`(`*`(`+`(f(`+`(a, h)), `-`(f(a)))), `*`(h)), h = 0);  

`+`(`-`(`/`(`*`(a), `*`(sqrt(`+`(`-`(`*`(`^`(a, 2))), 36)))))); (8.2.3)
 

 

 

with(Student[Calculus1]); -1 

MeanValueTheorem(f(x), x = -3 .. 5);  

Plot_2d
 

MeanValueTheorem(f(x), x = -3 .. 5, output = points);  

[`/`(`*`(`+`(`-`(`*`(3, `*`(sqrt(11)))), sqrt(3)), `*`(`+`(`-`(sqrt(11)), `*`(3, `*`(sqrt(3)))))), `*`(`+`(`*`(sqrt(11), `*`(sqrt(3))), `-`(17))))]; (8.2.4)
 

 

[1.3725]; (8.2.5)