SL M6 Calculus 2 - Integration Assignment 59 Marks

*B.Coulton* *1*

**Non-Calculator**

**1.** It is given that = *x*3+2*x* - 1 and that *y* = 13 when *x* = 2.

Find *y* in terms of *x*.

*Working:*

*Answer*:

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**(Total 6 marks)**

**2.** (a) Find d*x*.

**(4)**

(b) Find .

**(3)**

**(Total 7 marks)**

**Calculator Optional**

**3.** The acceleration, *a* m s-2, of a particle at time *t* seconds is given by

*a* = + 3sin 2*t*, for *t* Ã¢ÂÂ¥ 1.

The particle is at rest when *t* = 1.

Find the velocity of the particle when *t* = 5.

**(Total 7 marks)**

**4.** (a) Find the equation of the tangent line to the curve *y* = ln *x* at the point (e, 1), and verify that the origin is on this line.

**(4)**

(b) Show that (*x* ln *x* - *x*) = ln *x*.

**(2)**

(c) The diagram shows the region enclosed by the curve y = ln *x*, the tangent line in part (a), and the line *y* = 0.

Use the result of part (b) to show that the area of this region is e - 1.

**(4)**

**(Total 10 marks)**

**Calculator**

**5.** *In this question s represents displacement in metres and t represents time in seconds.*

The velocity *v* m s-1 of a moving body is given by *v* = 40 - *at* where *a* is a non-zero constant.

(a) (i) If *s* = 100 when *t* = 0, find an expression for *s* in terms of *a* and *t*.

(ii) If *s* = 0 when *t* = 0, write down an expression for *s* in terms of *a* and *t*.

**(6)**

Trains approaching a station start to slow down when they pass...