Term 1 Calculus - Limits and Derivatives

Evaluate

 $\lim_{x\rightarrow2}\ \frac{x^2-2}{4-x^2}$  

• -1

•  $-\frac{1}{2}$  

• 0

• does not exist

If

 $y\ =\ \sin^3\left(1-2x\right),\ then\ \frac{\text{d}y}{\text{d}x}\ is$  

•  $3\sin^2\left(1-2x\right)$  

•  $-2\cos^3\left(1-2x\right)$  

•  $-6\sin^2\left(1-2x\right)$  

•  $-6\sin^2\left(1-2x\right)\cos\left(1-2x\right)$  

Evaluate

 $\lim_{x\rightarrow\infty}\ \frac{\sqrt{x}-4}{4-3\sqrt{x}}$  

•  $-\frac{1}{3}$  

•  $-1$  

•  $0$  

•  $\infty$  

A point moves along the curve

 $y=x^2+1$  so that the x-coordinate is increasing at the constant rate of  $\frac{3}{2}$  units per second.  The rate, in units per second, at which the distance from the origin is changing when the point has the coordinate (1, 2) is equal to 

•  $\frac{7\sqrt{5}}{10}$  

•  $\frac{3\sqrt{5}}{2}$  

•  $\frac{3\sqrt{5}}{5}$  

•  $\frac{5}{2}$  

 $\lim_{h\rightarrow0}\ \frac{\sqrt{25+h}-5}{h}$  

Evaluate

•  $0$  

•  $\frac{1}{10}$  

•  $1$  

• Does not exist

Given the graph of  $f'$ , which could be a graph of  $f?$  

• I only

• II only

• III only

• I and III

The graph shows the velocity of an object on a 6-second interval. For how many values of  $t$   in the interval  $0<t<6\$  is the acceleration undefined?

• none

• one

• two

• three

The graph shows the velocity of an object on a 6-second interval. During what time interval (in sec) is the speed increasing?

• $0<t<3$

• $3<t<5$

• $5<t<6$

• never

The curve of $y=\frac{2x^2}{4-x^2}$  has

• two horizontal asymptotes and one vertical asymptote

• two vertical asymptotes and one slant asymptote

• one horizontal and one vertical asymptote

• one horizontal and two vertical asymptotes

Given the piece-wise function, which statement is true?

•  $f$  is continuous everywhere

•  $f$  is discontinuous only at  $x\ =\ 1.$  

•  $f$  is discontinuous at  $x=-2$  and at  $x=1$  

• If  $f\left(-2\right)$  is defined to be  $4$  , then  $f$  will be continuous everywhere.