Practice Test
(Midterm 2)
1. Sketch the graph of the following functions.
1
(a) y = x +
x
x
(b) y = 2
x −1
x2
(c) y = 2
x +1
(d) y = x3
2. Find the derivative of the following functions.
1
t2 + 3 4
√
1−t
p
(b) y = ln(x3 x2 + 1)
p
(c) y = ln(3x + 1)
(a) y = ln
(d) y = log2 (1 − x − x2 )
1
(e) y =
ln x
1 + ln x
(f) y =
x2
x2
(g) y = x
(h) y = e3 + eln x
(i) y =
e2x
2
+
2x
e
2
2
(j) y = e2x ln(4x)
(k) y =
ex − e−x
ex + e−x
(l) y = 4(x
2
+1)
(m) y = (e3x + 4)10
2
(n) y = 4(ex )3 − 4ex
dy
using the method of Implicit Differentiation.
dx
(a) x4 + 2x3 y 2 = x − y 3
p
(b) 2x + 2y = x2 + y 2
3. Find
(c) x + 2xy = 2. Also find the equation of tangent line at x = 1.
(d) x ln y + 2xy = 2. Also find the slope of the tangent at (1, 1).
(e) x2 y = ex+y
(f) yex = y 2 + x − 2. Also find the slope of tangents at x = 0.
(g) x2 + 4y 2 − 4x − 4 = 0. Also find the points where the curve has horizontal tangents and vertical
tangents.
4. Problem No. 8, 10, 11, 13, 16, on Page No. 795 − 796 of the text book. Problem No. 8 on Page No. 805.
5. Evaluate the following indefinite integrals.
Z
(a)
x(x − 1)2 dx
Z
2
(b)
xe(1+x ) dx
Z
7x3
√
(c)
dx
1 − x4
Z
3x
dx
−1
Z 1
(e)
x−
dx
(x + 1)2
Z
(f)
x(x2 − 1)10 dx
(d)
x2
Z
(g)
Z
(h)
Z
(i)
Z
(j)
x3 − 3x + 1
dx
x−1
(k)
(x3 − e3x ) dx
(l)
Z
Z
Z
3
y 2 ey dy
√
3
(m)
x2
dx
x3 − 4
(5x3 + 1)2 · 45x2
dx
(5x3 + 1)3
e(4x
2
−3)2
x5
e(2−3x6 )
Z
(n)
· 2(4x2 − 3) · 8x dx
dx
1600e0.4x dx
6. Suppose the rate of growth of the population of a city is predicted to be
dp
= 200t1.04
dt
where p is the population and t is the number of years past 2005. If the population in the year 2005 is
50, 000, what is the predicted population in the year 2015?
7. Suppose that the marginal cost for x units of a product is M C = 4x + 50, the marginal revenue is
M R = 500, and the cost of the production and sale of 10 units is $1000. What is the profit function for
this product?
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