007A Sample Midterm 1

This is a sample, and is meant to represent the material usually covered in Math 7A for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

Problem 1

Find the following limits:

(a) Find  ${\displaystyle \lim _{x\rightarrow 2}g(x),}$  provided that  ${\displaystyle \lim _{x\rightarrow 2}{\bigg [}{\frac {4-g(x)}{x}}{\bigg ]}=5.}$

(b) Find  ${\displaystyle \lim _{x\rightarrow 0}{\frac {\sin(4x)}{5x}}}$

(c) Evaluate  ${\displaystyle \lim _{x\rightarrow -3^{+}}{\frac {x}{x^{2}-9}}}$

Problem 2

Consider the following function  ${\displaystyle f:}$

${\displaystyle f(x)=\left\{{\begin{array}{lr}x^{2}&{\text{if }}x<1\\{\sqrt {x}}&{\text{if }}x\geq 1\end{array}}\right.}$

(a) Find  ${\displaystyle \lim _{x\rightarrow 1^{-}}f(x).}$

(b) Find  ${\displaystyle \lim _{x\rightarrow 1^{+}}f(x).}$

(c) Find  ${\displaystyle \lim _{x\rightarrow 1}f(x).}$

(d) Is  ${\displaystyle f}$  continuous at  ${\displaystyle x=1?}$  Briefly explain.

Problem 3

Let  ${\displaystyle y=2x^{2}-3x+1.}$

(a) Use the definition of the derivative to compute   ${\displaystyle {\frac {dy}{dx}}.}$

(b) Find the equation of the tangent line to  ${\displaystyle y=2x^{2}-3x+1}$  at  ${\displaystyle (2,3).}$

Problem 4

Find the derivatives of the following functions. Do not simplify.

(a)   ${\displaystyle f(x)={\sqrt {x}}(x^{2}+2)}$

(b)   ${\displaystyle g(x)={\frac {x+3}{x^{\frac {3}{2}}+2}}}$ where ${\displaystyle x>0}$

(c)   ${\displaystyle h(x)={\sqrt {x+{\sqrt {x}}}}}$

Problem 5

To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula:

${\displaystyle {\text{BSA}}={\frac {\sqrt {hm}}{60}}}$

where  ${\displaystyle h}$  is the height in centimeters and  ${\displaystyle m}$  is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass of  ${\displaystyle m=85.}$  What is the rate at  ${\displaystyle h=170}$  and  ${\displaystyle h=190?}$  Express your results in the correct units. Does the BSA increase more rapidly with respect to height at lower or higher heights?