Honors AP Calculus / Mr. Hansen |
Name: ______________________ |

No partial credit; no need to show work. Scoring is 2 points for each correct answer, –½ point for each incorrect answer, and 0 points for each omission.

1. If *f* ¢ ¢ (*x*) exists for all but finitely many *x* values in the interval *a* < *x* < *b*, then which of the following must be true?

I. Function *f* has a point of inflection at each value of *x* for which *f* ¢ ¢ (*x*) fails to exist.

II. Function *f* has either a vertical asymptote or a cusp at each value of *x* for which *f* ¢ ¢ (*x*) fails to exist.

III. Function *f* is continuous on at least one interval (*c*, *d*) where *a* £ *c* < *d* £ *b*.

A) I only

B) II only

C) III only

D) I and III only

E) II and III only

2. Which of the following is true if and only if function *g* has a limit of –¥ as *x* ® +¥ ?

A) If we choose *x* sufficiently large, *g*(*x*) will be negative and | *g*(*x*) | will be arbitrarily large.

B) If we choose *x* arbitrarily large, *g*(*x*) will be less than –10^{99}.

C) If we choose *x* sufficiently small, *g*(*x*) will be positive and | *g*(*x*) | will be arbitrarily large.

D) " e > 0, $ *M* > 0 ' *x* > *M* Þ | *g*(*x*) – *M* | < e

E) " e > 0, $ *M* > 0 ' *x* > *M* Þ | *g*(*x*) | > e

3. Which of the following, if true, would be sufficient to guarantee that function *f* has a point of inflection at *x* = 4.5?

I. *f* ¢ ¢ (4.499) < 0 and *f* ¢ ¢ (4.501) > 0, or vice versa

II. *x* < 4.5 Þ *f* ¢ ¢ (*x*) < 0, and *x* > 4.5 Þ *f* ¢ ¢ (*x*) > 0, or vice versa

III. *f* ¢ ¢ (4.5) = 0

IV. *f*(4.5) = 0

A) Condition III by itself is sufficient.

B) Conditions I and III, if both were true, would be sufficient.

C) Conditions I and IV, if both were true, would be sufficient.

D) Conditions II and III, if both were true, would be sufficient.

E) Conditions II and IV, if both were true, would be sufficient.

4. Please refer to the first 3 conditions (I through III) that were listed in question 3. Which of those conditions are *necessary* conditions (i.e., must be true) if function *f* has a point of inflection at *x* = 4.5?

A) I only

B) II only

C) III only

D) II and III only

E) I, II, and III

5. The signum function, sgn(*x*), is defined as |*x*|/*x* for all nonzero *x*, and sgn(0) = 0. Which of the following is an antiderivative of sgn(*x*) on Â \ {0}?

A) *x*^{4}/24

B) *x*^{3}/6

C) *x*^{3}/|6*x*|

D) |*x*|

E) none of these

6. A submarine is traveling due east. A sonar tracking sensor cannot detect the submarine at first, but at time *t* = 10 seconds, the submarine’s velocity is 30 m/sec. The submarine’s position is recorded as 4 km east of the sensor exactly 15 seconds later. The submarine undergoes acceleration of 0.06(*t* – 10)^{2} m/sec^{2}, where 10 £ *t* £ 30. Find, for the domain [10, 30], a function *s*(*t*) that expresses the submarine’s position in meters as a function of time *t* in seconds.

A) *s*(*t*) = 0.06(*t* – 10)^{4} + 30*t*

B) *s*(*t*) = 0.005(*t* – 10)^{4} + 30*t* + 2992.375

C) *s*(*t*) = 0.005(*t* – 10)^{4} + 30*t* + 2996.875

D) *s*(*t*) = 0.005(*t* – 10)^{4} + 30*t* + 3004.125

E) none of these

7. In engineering applications, quadrature is more widely used than definite integration based upon symbolic antidifferentiation because

A) computers cannot be programmed to perform symbolic antidifferentiation

B) most real-world functions do not have closed-form antiderivatives

C) quadrature is more accurate

D) antiderivatives cannot have cusps

E) antiderivatives cannot have points of discontinuity

8. Let *h*(*x*) = *f*(*x*) + *g*(*x*) for all real *x*. Which of the following statements must be true?

I. If *f* and *g* are both continuous except for one cusp each, then *h* has exactly one cusp.

II. If *f* and *g* are both continuous except for one cusp each, then *h* has exactly two cusps.

III. If *f* and *g* are both continuous, then *h* is continuous.

A) I only

B) II only

C) III only

D) I and III only

E) II and III only

9. Let *e* denote the base of the natural logarithm, approximately 2.718. All antiderivatives of *f*(*x*) = *e*^{2}*x*^{2} + *x* can be represented by the expression

A) *e*^{2}*x*^{3}/3 + *x*

B) *e*^{2}*x*^{3}/3 + *x*^{2}/2 + *C*, where *C* is an arbitrary constant

C) *e*^{3}*x*^{3}/3 + *x* + *C*, where *C* is an arbitrary constant

D) *e*^{3}*x*^{3}/3 + *x*^{2}/2

E)* e*^{3}*x*^{3}/3 + *x*^{2}/2 + *C*, where *C* is an arbitrary constant