Calculus 1 Science

Final Exam

Winter 1999 Semester

Numbers in square brackets [ ] are mark values

1. Evaluate the limits visually from the respective graphs. If the limit does not exist state in which

way (or "does not exist".)

*y=f(x)*

*y=g(x)*

*y=h(x)*

[3] a) (i) (ii) iii)

[2] b) Using the above functions, also find,

(i) (ii)

[1each] 2. Evaluate the following limits. If the limit does not exist state in which way (or "does not exist".)

[2] 3. Evaluate . If you use a calculator, make sure it is in radian mode.

[2] 4.a) State the conditions under which a function *f *(*x*)* *is continuous at *x=c*.

[2] b) Is *g*(*x*) continuous at *x=*1? Justify your answer.

[1] 5. a) State the limit definition of the derivative.

[3] b) Use the limit definition of the derivative to find *f '*(*x*) if .

[3 each] 6. Find for each of the following functions. Do not simplify.

[3] 7. Find if .

[4] 8. Find the equation of the tangent line to the curve at *x*=1.

[4] 9. Given , find *y''*.

[5] 10. Sketch the graph that satisfies these conditions.

* *when

* *when

* *when

* *when

* *when

[6]** **11. Sketch the following graph showing all steps.

, ,

[5]

[5] 13. A ball is thrown straight upward from the earth's surface with an initial velocity of 100 ft/sec (30 meters/sec.). The position of the ball after t seconds is given by feet.

- Determine the average velocity of the ball between 2 and 3 seconds.
- Determine the velocity function.
- What was the instantaneous velocity of the ball exactly 2.5 seconds after it was thrown?

[5] 14. Find the absolute extrema (maximum and minimum) of the function on the closed interval [-3,0].

[3each] 15. Evaluate the following**.**

[4] 16. Given , find *y* if *y=*45 when *x=*3.

[4] 17. Find the area of the region bounded by the graph of , the x axis, and the lines *x*=0 and *x*=5.