Application Center - Maplesoft

App Preview:

Calculus I: Lesson 5: Tangent Lines and Differentiability

You can switch back to the summary page by clicking here.

Learn about Maple
Download Application


 

L5-TangentLines.mws

Calculus I

Lesson 5: Tangent Lines and Differentiability

Example 1
For each of the following functions, we draw the graph of f(x) along with a secant and tangent line for x = a.

a) f(x) = cos(x)

> restart:

> a := 1;

a := 1

> f := x->cos(x);

f := cos

> h := 2.2;
q:= (f(a+h) - f(a))/h;
secline:= x -> f(a) + q*(x - a);
m:= D(f)(a);
tanline:= x -> f(a) + m*(x - a);

h := 2.2

q := -.4537703526-.4545454545*cos(1)

secline := proc (x) options operator, arrow; f(a)+q...

m := -sin(1)

tanline := proc (x) options operator, arrow; f(a)+m...

> plot([tanline(x), f(x), secline(x)], x=-5..5, color=[blue,green,magenta]);

[Maple Plot]

b) f(x) = x^4-1 and a = 3.

> f:= x -> x^4 -1;

f := proc (x) options operator, arrow; x^4-1 end pr...

> a:= 3;

a := 3

> h:= 2;

h := 2

> q:= (f(a+h) - f(a))/h;
secline:= x -> f(a) + q*(x - a);
m:= D(f)(a);
tanline:= x -> f(a) + m*(x - a);

q := 272

secline := proc (x) options operator, arrow; f(a)+q...

m := 108

tanline := proc (x) options operator, arrow; f(a)+m...

> plot({f(x), secline(x), tanline(x)}, x = a-1..a+3, color=[blue,red,magenta]);

[Maple Plot]

c) f(x) = sqrt(x) and a = 5.

> f:= x -> sqrt(x) ;

f := sqrt

> a:=5;

a := 5

> h:=7;

h := 7

> q:= (f(a+h) - f(a))/h; secline:= x -> f(a) + q*(x - a); m:= D(f)(a); tanline:= x -> f(a) + m*(x - a);

q := 2/7*sqrt(3)-1/7*sqrt(5)

secline := proc (x) options operator, arrow; f(a)+q...

m := 1/10*sqrt(5)

tanline := proc (x) options operator, arrow; f(a)+m...

> plot({f(x), secline(x), tanline(x)}, x= a-4..a+8, color=[blue,violet,magenta]);

[Maple Plot]

Example 2
For each of the following functions, plot f(x) and Df(x) on the same axes.

a) f(x) = x^3-7*x^2+12*x .

> f:= x -> x^3 - 7* x^2 + 12*x;

f := proc (x) options operator, arrow; x^3-7*x^2+12...

> D(f);

proc (x) options operator, arrow; 3*x^2-14*x+12 end...

> plot({D(f)(x), f(x)}, x = 0..5, color=[magenta,brown]);

[Maple Plot]

b) f(x) = |x|.

> f:= x -> abs(x);

f := abs

> D(f);

proc (a) options operator, arrow; abs(1,a) end proc...

> plot({D(f)(x), f(x)}, x = -4..4, color=[magenta,brown]);

[Maple Plot]

Example 3
Let f(x) = 2*sin(x)-cos(3*x) ; where x is from [-14,14].

a) Plot f(x) for x in [-14,1 4]. What is the period of f(x)?

> restart:

> f:= x -> 2 *sin(x) - cos(3*x);

f := proc (x) options operator, arrow; 2*sin(x)-cos...

> plot(f(x), x = -14..14);

[Maple Plot]

Period is 2*Pi .

b) Since the period of f(x) is 2*Pi , plot f(x) and D(f) on the same axes

for x in [-1,7].

> D(f);

proc (x) options operator, arrow; 2*cos(x)+3*sin(3*...

> plot({f(x), D(f)(x)}, x = -.1..7, color=[magenta,brown]);

[Maple Plot]

c) When f(x) has a max or min at x = a, what is D(f)(a)?

d) When f(x) is increasing (decreasing) what is the sign of D(f)?

When f is increasing D(f) >0, and when f is decreasing D(f) <0.

Example 4
Let f(x) = x*sin(1/x) if x > 0 and 0 if x <= 0 .

Plot f(x). Use the graph to determine whether D(f)(0) exists.

> f:= x -> piecewise( x > 0, x*sin(1/x), 0);

f := proc (x) options operator, arrow; piecewise(0 ...

> plot(f(x), x = -0.1..0.1);

[Maple Plot]

> plot(f(x), x = -.0001...0001);

[Maple Plot]

Based on the plots, we conjecture that f is not differentiable when x = 0.

Example 5
Let f(x) = x^2*sin(1/x) for x > 0 and 0 for x <= 0.

Plot f(x). Use the graph to determine if D(f)(0) exits.

> f:= x -> piecewise(x>0, x^2 * sin(1/x), 0);

f := proc (x) options operator, arrow; piecewise(0 ...

> plot(f(x), x = -.00001...00001);

[Maple Plot]

Based on the plot, we conjecture that D(f)(0) = 0.

Example 6
Let f(x) = 3-2*x^2 . Plot f(x), the tangent line when x = .5 and

the normal line when x =.5 on the same axes.

> restart:

> f:=x -> 3 - 2*x^2;

f := proc (x) options operator, arrow; 3-2*x^2 end ...

> D(f)(.5);

-2.0

> f(.5);

2.50

> t:= x -> -2*(x - .5) + 2.5;

t := proc (x) options operator, arrow; -2*x+3.5 end...

> n:= x -> (1/2)*(x - .5) + 2.5;

n := proc (x) options operator, arrow; 1/2*x+2.2500...

> plot({f(x), t(x), n(x)}, x = -1..1, color=[magenta,brown, wheat]);

[Maple Plot]

Example 7
Let f(x) = |cos(x)|. Determine where f(x) is not differentaible.

> restart:

> f:= x -> abs(cos(x));

f := proc (x) options operator, arrow; abs(cos(x)) ...

> plot(f(x), x = -7..7);

[Maple Plot]

From the plot we conjecture that f(x) is not differentiable at multiples of Pi/2 and 3*Pi/2 .