MATH 151 Engineering Math I, Spring 2014
JD Kim
Week5 Section 3.2, 3.3
Section 3.2 Differentiation Formulas.
Differentiation Formulas
1. Constant rule: If f (x) = c, where c is a constant then f ′ (x) = 0.
2. Power rule: If f (x) = xn , then f ′ (x) = n · xn−1 .
3. Constant times a function rule:
d
d
cf (x) = (cf (x))′ = c f (x) = cf ′ (x).
dx
dx
4. Sum/Difference rule: If f (x) = g(x) ± h(x), then f ′ (x) = g ′(x) ± h′ (x).
5. Product rule: If f (x) = g(x) · h(x), then f ′ (x) = g ′ (x)h(x) + g(x)h′ (x).
6. Quotient rule: If f (x) =
g ′(x)h(x) − g(x)h′ (x)
g(x)
, then f ′ (x) =
.
h(x)
(h(x))2
Ex1) Find the derivative of the following functions.
1-1) g(x) = x5 + 8x2 − 16x + 2 − π 2
1-2) f (t) = (1 −
√
t)2
1
1-3) H(s) =
s 5
2
√
x − 3x x
√
1-4) F (x) =
x
1-5) y = (x3 − x2 − 2x + 1)(5x4 − 20x3 + 5x + 3)
1-6) f (u) =
1 − u2
1 + u2
Ex2) If f (2) = 1, f ′ (2) = 6, g(2) = −3, and g ′ (2) = 2, find the value of (f g)′(2).
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Ex3) Find the equation of the tangent line to the graph of f (x) = x +
point (1, 2).
√
x at the
√
Ex4) At what point on the curve y = x x is the tangent line parallel to the line
3x − y + 6 = 0.
3
Ex5) Show there are two tangent lines to the parabola y = x2 that pass through
the point (0, −4). Find the equations of these lines.
4
Ex6) Sketch the graph of f (x) and f ′ (x)




−1 − 2x




f (x) =
x2






 x
5
on the same axis.
if x < −1
if − 1 ≤ x < 1
if x ≥ 1
Ex7) If f (x) =



 x2


 mx + b
if x ≤ 2
, find the value of m and b that make f (x)
if x > 2
differentiable everywhere.
6
~ =< t2 + 2t, t3 + 3t2 > is the position of a moving object at time t,
Ex8) If r(t)
where the position is measured in feet and the time in seconds, find the velocity and
speed at time t = 1.
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Section 3.3 Rates of change in the Natural and
Social Sciences.
Ex9) A particle moves according to the equation of motion s(t) = 4t3 −9t2 +6t+2,
where s(t) is measured in meters and t in seconds.
9-1) Find the velocity at time t.
9-2) When is the particle at rest?
9-3) When is the particle moving in the positive direction?
9-4) Draw a diagram that represents the motion of the particle.
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9-5) Find the distance traveled in the first 3 second.
Ex10) A ball is thrown vertically upward with a velocity of 80 feet per second.
The height after t seconds is given by h(t) = 80t − 16t2 . What is the maximum
height of the ball.
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