1. Use the Intermediate Value Theorem to show that the following equation has
solution for 0 x 1.
e x x 2 2 cos(2 x ) 1
2. Suppose that f is a differentiable function and that f (0) 3 , and f ( x ) 5 for
all value of x.
1. Explain in your own words what it means to say that
lim f ( x ) 3 and lim f ( x ) 7
In this situation, is it possible that lim f ( x ) exists? Explain.
2. Sketch the graph of the following function and u
1. Using the graph of the function g ( x ) , sketch a graph of the shifted or scaled
function, say which kind of shift or scale it is, and compare with the original
(a) g ( x / 3)
(b) g ( x 1)
2. Determine whe
1. Use the tangent line approximation to evaluate ln( 0.98) in two ways. First,
find the tangent line to the whole function using the chain rule. Second, break the
calculation into two pieces by writing the function a
1. For the following figure, label
(a) One critical point
(b) One point with a positive derivative
(c) One point with a negative derivative
(d) One point with a positive second derivative
(e) One point with a negative s
by implicit differentiation.
(a) x 2 xy y 2 4
(b) sin( x y ) y 2 cos x
(c) e y x y
(d) tan 1 ( x 2 y ) x xy 2
2. Show that
x x2 1
3. f ( x) 2 x e x , x R , find ( f 1 ) '(1) .
4. f ( x
1. Suppose a population of bacterial doubles every hour, but that 1106 individuals
are removed before reproduction to be converted into valuable biological byproducts. Suppose the population begins with b0 3 106 bacter
1. Find the derivative of the following polynomial function. State where you used
the sum, constant product, and power rules.
x2 x3 x4
2. Compute the derivative of the following function in the two ways given.
1. Sketch the graph of the function y
e 1 and determine its domain and
2. Find the domain of each function.
(a) f ( x)
1 e1 x
(b) g (t ) sin(e t )
3. Sketch the graphs of f ( x) 1 x and its inver
1. Are inverse trigonometric functions such as arcsin, arccos, and arctan functions
are continuous? Explain why.
f ( x) e x
if x 0
if 0 x 1
if x 1
(a) Sketch the graph of f.
(b) Find the numbers at which f i