MATH1220: Midterm 1 Practice Problems
The following are practice problems for the first exam.
1. Compute the following derivatives:
p
(a) Dx ln 3x2 + 2
(b) Dx esin x
d loga (2x2 ) cos x
(c)
dx
d h 3x2 +x+1 i
4
(d)
dx
2. Compute the following integrals:
Z
6x2 + 16x
(a)
dx
x3 + 4x2 − 3
Z 5
2
(b)
5xex dx
Z0
(c)
3x dx
Z
(d)
tan x dx
5x − 3
and verify that it is actually the inverse by
2x − 1
−1
−1
showing that f ◦ f (y) = y and f ◦ f (x) = x.
3. Find the inverse of the function f (x) =
4. Show that f (x) = x5 + 2x3 + 4x + sin(πx) has an inverse (don’t try to find the inverse) and
compute (f −1 )0 (7). (Hint: You can find an x such that f (x) = 7 by inspection)
5. Compute
d (1 + x2 )cos x
dx
6. A radioactive substance loses 15% of its radioactivity in 2 days. What is its half-life?
7. Find the general solution to the following differential equation:
dy
2y
+
= (x + 1)3
dx x + 1
8. Use Euler’s Method with h = 0.5 to approximate the solution to
y 0 = 2y − 2x
y(0) = 1
over the interval [0, 1].
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9. Sketch the solution to y 0 = x2 − y, whose slope field is shown below, satisfying the initial
conditions y(−2) = 1.
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