MATH 151 Engineering Mathematics I
Week In Review
Fall, 2015, Problem Set 7 (Exam2 Review)
JoungDong Kim
1. Find f ′ (x).
(a) f (x) = x3 e2x
(b) f (x) =
(x3 + 1)
(x2 + 1)2
1
4
2. Let f (x) = x 3 , find the x-coordinate of the point on the graph of f where the tangent is perpendicular to the line x + 4y = 1.
3. For what value(s) of x does the graph of f (x) = x + 2 sin x have a horizontal tangent for 0 ≤ x ≤
2π?
2
4. Find the x-coordinate of tangent points of tangent lines to the parabola y = x2 that pass through
the point (0, −4).
5. Let f be a differentiable function, and let G be the function defined by G(x) = f (x)[f (f (x))],
compute the derivative of G with respect to x.
3
sin(7x)
.
x→0 x cos(8x)
6. Evaluate lim
7. If f (x) = sin(g(x)), find f ′ (2) given that g(2) =
4
π
π
and g ′(2) = .
3
4
8. Find
dy
dx
(a) 2xy + π sin(y) = 2π
(b) sin(xy) + x2 sin(y 2) = 0
9. Find the slope of tangent line to the curve x2 + y 2 = 13 at the point (3, −2).
5
10. A particle moves according to the equation s(t) = t2 − t, t ≥ 0, where t is measured in seconds
and s is in feet. What is the total distance the particle travels during the first 2 seconds?
11. Find a tangent and acceleration vectors for r(t) = ht sin t, cos(2t)i at t =
6
π
.
2
12. Given the curve parameterized by x = 2t3 , y = sin(πt), find the slope of line tangent to the curve
at the point (2, 0).
13. Find the point(s) on the curve x = t2 + 4t, y = t2 + 2t where the tangent line is vertical and
horizontal.
7
14. The 2015th derivative of f (x) =
1
.
(1 − x)2
15. The 2015th derivative of g(x) = sin(2x).
8
16. A water tank has the slape of an inverted circular cone with radius 5m and height 7m. A solution
is being poured into the tank in such a way that the height of the fluid is increasing at a rate of
1
m per minute. At what rate is the volume increasing when the height is 4m?
2
9
17. Find the linear and quadratic approximations for f (x) = cos(2x) at a =
18. Use the linear approximation to estimate the (2.01)4 .
10
π
.
2
19. Find the limit.
(a) lim (0.3)−x
x→∞
(b) lim (0.3)−x
x→−∞
x
2−x
1
(c) lim+
x→2
4
x
2−x
1
(d) lim−
x→2
4
2
(e) lim− e x−1
x→1
2
(f) lim+ e x−1
x→1
e3x − e−3x
x→∞ e3x + e−3x
(g) lim
e3x − e−3x
(h) lim 3x
x→−∞ e
+ e−3x
11
2
20. Let f (x) = ex , find f ′ (x).
12