Download 9 fxx = ( ) 2

Math 125-06 [ Review problems for Quiz #2 (Spring 2013)]
1. Find the slope of the tangent line to the graph of the function at the given point.
f ( x) = –5 x 2 +10, (–2, –10)
2. Use the limit definition to find the slope of the tangent line to the graph of
the point (5, 7).
f ′( x) = lim ∆x→0
f ( x) = 4 x + 29 at
f ( x + ∆x ) − f ( x )
∆x
3. Find the derivative of the following function using the limiting process.
f ( x) = 9 x – 6
4. Find the derivative of the following function using the limiting process.
f ( x) =
2
x–9
5. Find the derivative of the function.
f ( x) = 2 x 3 – 3x 2 +1
6. Find the derivative of the function.
h( x) = 15 x 23 + 11x13 − 4 x10 + 3x − 7
7. Find the derivative of the function.
f ( x) =
1
x3
8. Find the marginal profit for producing x units. (The profit is measured in dollars.)
P = −2 x 2 + 72 x − 145
9. The profit (in dollars) from selling x units of calculus textbooks is given by
P = − 0.05 x 2 + 20 x − 3000 . Find the additional profit when the sales increase from 145 to 146
units. Round your answer to two decimal places.
10. When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold.
When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function
is linear and that the marginal and fixed costs are $0.10 and $ 25, respectively. Find the marginal
profit when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold.
11. Use the product Rule to find the derivative of the function
12. Find the derivative of the function.
f ( x) = x8 (7 + 6 x) 4
f ( x ) = x ( x 2 + 3) .
13. Find the derivative of the function
function.
 x+5 
g ( x) =  2

 x +5
5
14. Find the third derivative of the function
f ( x ) = x5 − 3x 4 .
15. Use the graph of y = f ( x ) to identify at which of the indicated points the derivative f '( x )
changes from positive to negative.
16. Identify the open intervals where the function
f ( x) = 4 x 2 − 3 x + 2 is increasing or decreasing.
17. For the given function, find all critical numbers.
y = x3 − 9 x 2 − 48 x + 4
18. A fast-food
food restaurant determines the cost model, C ( x ) = 0.3 x + 4500,
revenue model, R ( x) =
0 ≤ x ≤ 30000 and
45000 x − x
, for 0 ≤ x ≤ 30000 where x is the number of
20000
2
hamburgers sold. Determine
mine the intervals on which the profit function is increasing and on
which it is decreasing.
19. For the function
f ( x) = 4 x3 − 36 x 2 + 1 :
(a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.
20. Locate the absolute extrema of the function
[ –2, 2] .
f ( x) = −3 x 2 − 6 x + 2 on the closed interval
21. Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose
is taken can be modeled by
x( x) =
4000t
, t > 0 . Find the t-value at which x is maximum.
t2 + 5
Round your answer to two decimal places.
22. Locate the absolute extrema of the given function on the closed interval [–36,36].
[
f ( x) =
36 x
x + 36
2