MATH1220: Midterm 1 Practice Problems
The following are practice problems for the first exam.
1. Compute the following derivatives:
p
3x
(a) Dx ln 3x2 + 2 = 2
3x + 2
(b) Dx esin x = esin x cos x
2 cos x
d (c)
loga (2x2 ) cos x =
− sin x loga 2x2
dx
x ln a
2
d h 3x2 +x+1 i
= 43x +x+1 (6x + 1) ln 4
4
(d)
dx
2. Compute the following integrals:
Z
6x2 + 16x
(a)
dx = 2 ln(x3 + 4x2 − 3) + C
x3 + 4x2 − 3
Z 5
2
5 25
e −1
(b)
5xex dx =
2
Z0
3x
x
+C
(c)
3 dx =
ln 3
Z
(d)
tan x dx = − ln | cos x| + C
5x − 3
and verify that it is actually the inverse by
2x − 1
−x + 3
showing that f ◦ f −1 (y) = y and f −1 ◦ f (x) = x. Answer: f −1 (x) =
−2x + 5
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) Answer:
1
(f −1 )0 (7) =
15 − π
2x cos x
d 2 cos x
2 cos x
2
(1 + x )
= (1 + x )
− sin x ln(1 + x ) +
5. Compute
dx
1 + x2
6. A radioactive substance loses 15% of its radioactivity in 2 days. What is its half-life? Answer:
2 ln 0.5
τ1/2 =
ln 0.85
7. Find the general solution to the following differential equation:
dy
2y
+
= (x + 1)3
dx x + 1
Answer:
y=
(x + 1)4
C
+
6
(x + 1)2
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]. Answer: y(1) ≈ 3.5
1
9. Sketch the solution to y 0 = x2 − y, whose slope field is shown below, satisfying the initial
conditions y(−2) = 1.
2