Mr. Orchard’s Math 142 WIR
4.4, 4.5, 5.1
1. Differentiate the following functions:
(a) e2x
(b) ln(2x2 − 13x)
(c) 5x
3 −12x2
(d) log4 (x2 )
(e) x3 84x +
log3 x
3x
Week 7
Mr. Orchard’s Math 142 WIR
2. G(x) = log3
x2 (x2 +5)4
(x3 −4)5
4.4, 4.5, 5.1
Week 7
(a) Use the properties of logarithms to rewrite G(x) in terms of simpler logarithms.
(b) Find G0 (x)
3. Find where the following functions have horizontal tangent lines:
(a) 43x
2 −24x
(b) (−12x − 21)e4x+5
Mr. Orchard’s Math 142 WIR
4.4, 4.5, 5.1
Week 7
7
4. Find the equation for the tangent line of f (x) = ln(xex ) at the point (1, 1).
5. $2,600 is invested in a savings account with a nominal yearly rate of 3.5% compounded
continuously. At what rate is the balance growing after 2 years?
Mr. Orchard’s Math 142 WIR
4.4, 4.5, 5.1
6. A certain product has the price-demand function x =
per unit and x is the quantity demanded.
Week 7
√
168 − 2p, where p is the price
(a) Obtain a formula for the elasticity of demand.
(b) Is the demand elastic, inelastic, or unit elasticity at p = $12?
(c) Find the price so that the revenue is maximized.
(d) What is the revenue function for this product?
(e) What is the maximum revenue for the company?
Mr. Orchard’s Math 142 WIR
4.4, 4.5, 5.1
Week 7
7. For a particular calculator, the price is $73 and E(73) = 0.33. If the price is decreased
by $11, what is the approximate change in demand?
8. Below is the graph of f (x). Use it to answer the following:
4
3
2
1
0
-1
-2
-3
-4
-3
-2
(a) Find the critical values of f (x).
(b) Find the intervals where f (x) is increasing.
-1
0
1
Mr. Orchard’s Math 142 WIR
4.4, 4.5, 5.1
Week 7
9. Find the critical values for f (x) = 4x4 − 72x2 + 12.
10. Find the intervals where g(x) = 4x8 + 32x7 + 12 is increasing.
11. The domain of f (x) is all real numbers and f 0 (x) = (x + 1)4 (x − 5)3 (x − 6)5 e(x−11) .
(a) Find the critical values of f (x).
(b) Find the intervals where f (x) is increasing.
(c) Find the intervals where f (x) is decreasing.
Mr. Orchard’s Math 142 WIR
12. g(x) =
4.4, 4.5, 5.1
Week 7
x4
.
x+24
(a) Find the critical values of g(x).
(b) Find the intervals on which g(x) is increasing.
(c) Find the intervals on which g(x) is decreasing.
(d) Find the x-coordinate(s) of the relative extrema of g(x). Indicate which are minima
and which are maxima.
Mr. Orchard’s Math 142 WIR
4.4, 4.5, 5.1
Week 7
13. The domain of f (x) is (−∞, ∞). Below is the graph of f 0 (x). Use it to answer the
following:
10
8
6
4
2
0
-2
-4
-2
-1
0
(a) Find the critical values of f (x).
(b) Find the intervals where f (x) is decreasing.
1
2
3