Randolph College
Department of Mathematics and Computer Science

 

Homework # 10 - Math 149


The problems on this homework must be done alone.  The honor code is in effect.

 First name:   Last name:


 

1.  On what line is the first error in the following? Let \(f(x) = 4x^2\).

$$\begin{array}{rclr} f'(x) & = & \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\hspace{.5 in} & \mbox{Line 1}\\ & = & \lim_{h \rightarrow 0} \frac{4x^2 + h - 4x^2}{h} & \mbox{Line 2}\\ & = & \lim_{h \rightarrow 0} \frac{h}{h} & \mbox{Line 3}\\ & = & \lim_{h \rightarrow 0} 1 & \mbox{Line 4}\\ & = & 1 & \mbox{Line 5} \end{array}.$$

a.  Line 1.

b.  Line 2.

c.  Line 3.

d.  Line 4.

e.  Line 5.

 

2.  On what line is the first error in the following? Let \(f(x) = x^3\).

$$\begin{array}{rclr} f'(x) & = & \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}\hspace{.5 in} & \mbox{Line 1}\\ & = & \lim_{h \rightarrow 0} \frac{(x+h)^3 - x^3}{h} & \mbox{Line 2}\\ & = & \lim_{h \rightarrow 0} \frac{x^3 + h^3 - x^3}{h} & \mbox{Line 3}\\ & = & \lim_{h \rightarrow 0} \frac{h^3}{h} & \mbox{Line 4}\\ & = & \lim_{h \rightarrow 0} h^2 & \mbox{Line 5}\\ & = & 0^2 & \mbox{Line 6}\\ & = & 0 & \mbox{Line 7} \end{array}.$$

a.  Line 1.

b.  Line 2.

c.  Line 3.

d.  Line 4.

e.  Line 5.

 

3.  Take the derivative of \(f(x) = \frac{1}{x^2}\) (Feel free to use the derivative rules).

a.  \(f'(x) = \frac{-2}{x}\)

b.  \(f'(x) = \frac{2}{x}\)

c.  \(f'(x) = \frac{2}{x^3}\)

d.  \(f'(x) = \frac{-2}{x^3}\)

e.  \(f'(x) = 0\)

 

4.  Take the derivative of \(f(x) = 3x^2 - 5x + 10\) (Feel free to use the derivative rules).

a.  \(f'(x) = 6x - 5\)

b.  \(f'(x) = 3x - 5\)

c.  \(f'(x) = 2x^2 - 5x\)

d.  \(f'(x) = 2x^2 - 5\)

e.  \(f'(x) = 2x^2 - 4x + 10\)

 

5.  Take the derivative of \(g(x) = 2^{-100}\) (Feel free to use the derivative rules).

a.  \(g'(x) = (100) 2^{101}\)

b.  \(g'(x) = (-100) 2^{-101}\)

c.  \(g'(x) = (100) 2^{99}\)

d.  \(g'(x) = (-100) 2^{-99}\)

e.  \(g'(x) = 0\)

 

6.   Which of the following limits represents the slope of the tangent line to the curve \(y = f(x) = x^2\) at \(x = 2\)?

a.  \(\lim_{h \rightarrow 0} \frac{ 2+h^2 - 2^2}{h}\)

b.  \(\lim_{h \rightarrow 0} \frac{ (2+h)^2 - 2^2}{h}\)

c.  \(\lim_{h \rightarrow 0} \frac{ 1+h^2 - 2^2}{h}\)

d.  \(\lim_{h \rightarrow 0} \frac{ (1+h)^2 - 1^2}{h}\)

e.  \(\lim_{h \rightarrow 0} \frac{ (1+h)^2 - h^2}{h}\)