MATH 151 Engineering Mathematics I

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MATH 151 Engineering Mathematics I

Fall 2017, WEEK 4

JoungDong Kim

Week 4

Section 2.8, 3.1, 3.2, The Derivative as a Function, Derivatives of Polynomial and Exponential Functions, The Product and Quotient Rules.

Section 2.8 The Derivative as a Function

In the preceding section we considered the derivative of a functon f at a fixed number a: f′

(a) = lim

h→₀

f (a + h) − f(a) h

Here we change our point of view and let the number a vary. If we replace a by a variable x, we obtain f′ (x) = lim h→₀ f (x + h) − f(x) h

Other Notation (Leibniz notation) f′ (x) = y′ = dy dx = df dx

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Ex.1) Find the derivative of the following functions. a) f (x) = x2_{− 8x + 9}

b) g(x) = x x + 1

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c) h(x) =√_{2x − 1}

d) k(x) = _√4 x

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Definition. Differentiable A function f is differentiable at a if f′

(a) exists. Theorem. If f is differentiable at a, then f is continuous at a.

How can a function FAIL to be differentiable

1. If the graph of a function f has a “corner” or “kink” in it, then the graph of f has no tangent at that point and f is not differentiable there.

2. If f is not continuous at a, then f is not differentiable at a.

3. The curve has a vertical tangent line when x = a, f is not differentiable at a.

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Ex.2) The graph of f is given. State, with reasons, the numbers at which f is not differentiable.

Ex.3) Where is f (x) = |x2_{− 4| not differentiable.}

Ex.4) Where is f (x) = |x + 1|

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Ex.5) Given the graph of f (x) below, sketch the graph of the derivative. a)

b)

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Section 3.1 Derivatives of Polynomials and Exponential

Func-tions

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) = nxn−₁

.

3. Constant times a function rule: (cf (x))′

= cf′

(x) or d

dx[cf (x)] = c d dxf (x). 4. Sum/Difference rule: If f (x) = g(x) ± h(x), then f′

(x) = g′

(x) ± h′

(x).

5. Derivative of the Natural Exponential Function: (ex

)′ = ex or d dxe x = ex .

Ex.6) Find the derivative of the following functions. a) g(x) = x5_{+ 8x}2_{− 16x + 2 − π}2 b) f (t) = (1 −√t)2 c) H(s) =s 2 5 d) F (x) = x − 3x √ x √ x

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Ex.7) Find the equation of the tangent line to the graph of f (x) = x +√x at the point (1, 2).

Ex.8) At what point on the curve y = x√_{x is the tangent line parallel to the line 3x − y + 6 = 0.}

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Ex.9) Show there are two tangent lines to the parabola y = x2 _{that pass through the point (0, −4). Find}

the equations of these lines.

Ex.10) Sketch the graph of f (x) and f′

(x) on the same axis. For what values of x is the function f (x) not differentiable. f (x) =                −1 − 2x if x < −1 x2 _{if − 1 ≤ x < 1} x _{if x ≥ 1}

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Ex.11) Find the parabola with equation y = ax2_{+ bx whose tangent line at (1, 1) has equation y = 3x−2.} Ex.12) If f (x) =        x2 _{if x ≤ 2} mx + b if x > 2

, find the value of m and b that make f (x) differentiable every-where.

Ex.13) If r(t) = ht2_{+ 2t, t}3_{+ 3t}2_{i is the position of a moving object at time 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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Higher Derivatives

If f is a differentiable function, then its derivative f′

is also a function, so f′

may have a derivative of its own, denoted by (f′

)′

= f′′

. This new function f′′

is called the second derivative of f because it is the derivative of the derivative of f . Using Leibniz notation, we write the second derivative of y = f (x) as d dx dy dx = d 2_y dx2

Similarly, the third derivative is the derivative of the second derivative, denoted by f′′′

(x). In general, the nth derivative of f (x) is denoted by f(n)_(x). Ex.14) If f (x) = x3_{− x, find f}′′ (x), f′′′ (x), and f(4)_.

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Section 3.2 The Product and Quotient Rules

The Product Rule

If f and g are both differentiable, then [f (x)g(x)]′

= f′

(x)g(x) + f (x)g′

(x)

The Quotient Rule If f and g are both differentiable, then f (x) g(x) ′ = f ′ (x)g(x) − f(x)g′ (x) (g(x))2

Ex.15) Find the derivative of the following functions. a) y = (x3_{− x}2_{− 2x + 1)(5x}4_{− 20x}3_{+ 5x + 3)}

b) f (x) = 1 − x

2

1 + x2

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c) f (x) = x

2_e

x2 _{+ e}x

Ex.16) If f (2) = 1, f′

(2) = 6, g(2) = −3, and g′

(2) = 2, find the values of (f g)′

(2) and (f /g)′

(2).

Ex.17) If f (x) =√xg(x), where g(4) = 2 and g′

(4) = 3, find f′

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Ex.18) Find the 25th derivative of f (x) = 1 x.

Ex.19) If f (x) = xex

, find the nth derivative, f(n)_(x).

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Ex.20) Find an equation of the tangent line to the curve y = e

(1 + x2₎ at the point (1, 1 2e).

Ex.21) Find equations of the tangent lines to the curve y = x − 1