Exercise 5 | Wiskunde op Tilburg University

Consider the functions

$y_1(x)=\frac{1}{3}x^3+bx^2-x+1$ and

$y_2(x)=bx^2+1$ .

Determine all the values of

$b$ such that the graphs of the functions are tangent.

Antwoord 1 correct

Correct

Antwoord 2 optie

The graphs of the functions are tangent for every value of

$b$ .

Antwoord 2 correct

Fout

Antwoord 3 optie

$b=1$ or

$b=-1$

Antwoord 3 correct

Fout

Antwoord 4 optie

$b=\sqrt{3}$ ,

$b=-\sqrt{3}$ or

$b=0$

Antwoord 4 correct

Fout

Antwoord 1 optie

There is no value of

$b$ such that the graphs of the functions are tangent.

Antwoord 1 feedback

Correct: The graphs of the functions are tangent for an

$x$ -waarde if the both function values are the same.

$y'_1(x)=x^2+2bx-1$ and

$y'_2(x)=2bx$ . Hence,

$y'_1(x)=y'_2(x)$ for

$x=1$ or

$x=-1$ .

However,

$y_1(x)=y_2(x)$ for

$x=0$ or

$x=\sqrt{3}$ or

$x=-\sqrt{3}$ .

Hence, there is no value of

$b$ such that the graphs of the functions are tangent.

Go on.

Antwoord 2 feedback

Wrong: Both the function values and the derivatives have to be equal.

Try again.

Antwoord 3 feedback

Wrong: Do not confuse

$b$ with

$x$ .

Try again.

Antwoord 4 feedback

Wrong: Do not confuse

$b$ with

$x$ .

Try again.