Plot the derivative of f (x) = x/(x2 + 1) over [4, 4]. Use the graph to determine the

Are the following statements true or false? If false, state the correct version. (a) f g denotes the function whose value at x is f (g(x)). (b) f/g denotes the function whose value at x is f (x)/g(x). (c) The derivative of the product is the product of the derivatives (d) (e)

Find (f/g)’ (1) if f (1) = f’(1) = g(1) = 2 and g’(1) = 4.

Find g(1) if f (1) = 0, f’(1) = 2, and (fg)’(1) = 10.

In Exercises 1–6, use the Product Rule to calculate the derivative.

In Exercises 1–6, use the Product Rule to calculate the derivative. ,

In Exercises 1–6, use the Product Rule to calculate the derivative. ,

In Exercises 7–12, use the Quotient Rule to calculate the derivative.

In Exercises 7–12, use the Quotient Rule to calculate the derivative.

In Exercises 7–12, use the Quotient Rule to calculate the derivative. ,

In Exercises 7–12, use the Quotient Rule to calculate the derivative. ,

In Exercises 7–12, use the Quotient Rule to calculate the derivative.

In Exercises 7–12, use the Quotient Rule to calculate the derivative.

In Exercises 13–16, calculate the derivative in two ways. First, use the Product or Quotient Rule; then rewrite the function algebraically and apply the Power Rule directly

In Exercises 1316, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and apply the Power Rule directly. x) = x2(3 + x1)

In Exercises 1316, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and apply the Power Rule directly. ) = t2 1 t 1

In Exercises 1316, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and apply the Power Rule directly. ) = x3 + 2x2 + 3x1 x

In Exercises 1738, calculate the derivative. f (x) = (x3 + 5)(x3 + x + 1)

In Exercises 1738, calculate the derivative. x) = (4ex x2)(x3 + 1)

In Exercises 1738, calculate the derivative. dx x=3 , y = 1 x + 10

In Exercises 1738, calculate the derivative. dx x=2 , z = x 3x2 + 1

In Exercises 1738, calculate the derivative. x) = ( x + 1)(x 1)

In Exercises 1738, calculate the derivative. x) = 9x5/2 2 x

In Exercises 1738, calculate the derivative. dx x=2 , y = x4 4 x2 5

In Exercises 1738, calculate the derivative. x) = x4 + ex x + 1

In Exercises 1738, calculate the derivative. dx x=1 , z = 1 x3 + 1

In Exercises 1738, calculate the derivative. x) = 3x3 x2 + 2 x

In Exercises 1738, calculate the derivative. ) = t (t + 1)(t2 + 1)

In Exercises 1738, calculate the derivative. x) = x3/2 2x4 3x + x1/2

In Exercises 1738, calculate the derivative. t) = 31/2 51/2

In Exercises 1738, calculate the derivative. ) = 2(x 1)

In Exercises 1738, calculate the derivative. x) = (x + 3)(x 1)(x 5)

In Exercises 1738, calculate the derivative. x) = ex (x2 + 1)(x + 4)

In Exercises 1738, calculate the derivative. x) = ex x + 1

In Exercises 1738, calculate the derivative. ) = ex+1 + ex e + 1

In Exercises 13–16, calculate the derivative in two ways. First use the Product or Quotient Rule; then rewrite the function algebraically and apply the Power Rule directly:

In Exercises 1738, calculate the derivative. x (ax + b)(abx2 + 1) (a, b constants)

In Exercises 1738, calculate the derivative. t xt 4 t2 x (x constant)

In Exercises 1738, calculate the derivative. x ax + b cx + d (a, b, c, d constants)

In Exercises 3942, calculate the derivative using the values: and

In Exercises 3942, calculate the derivative using the values: F (4), where F (x) = x2f (x)

In Exercises 3942, calculate the derivative using the values: G (4), where G(x) = (g(x))2

In Exercises 3942, calculate the derivative using the values: H (4), where H (x) = x g(x)f (x)

Calculate F (0), where F (x) = x9 + x8 + 4x5 7x x4 3x2 + 2x + 1 Hint: Do not calculate F (x). Instead, write F (x) = f (x)/g(x) and express F (0) directly in terms of f (0), f (0), g(0), g (0). 44. P

Proceed as in Exercise 43 to calculate F (0), where F (x) =  1 + x + x4/3 + x5/3 3x5 + 5x4 + 5x + 1 8x9 7x4 + 1 4

Use the Product Rule to calculate d dx e2x .

Plot the derivative of f (x) = x/(x2 + 1) over [4, 4]. Use the graph to determine the intervals on which f (x) > 0 and f (x) < 0. Then plot f and describe how the sign of f (x) is reflected in the graph of f . 47

Plot f (x) = x/(x2 1) (in a suitably bounded viewing box). Use the plot to determine whether f (x) is positive or negative on its domain {x : x = 1}. Then compute f (x) and confirm your conclusion algebraically. 4

Let P = V 2R/(R + r)2 as in Example 7. Calculate dP/dr, assuming that r is variable and R is constant.

Find a > 0 such that the tangent line to the graph of f (x) = x2ex at x = a passes through the origin (Figure 4). y x a f(x) = x2ex

Current I (amperes), voltage V (volts), and resistance R (ohms) in a circuit are related by Ohm's Law, . (a) Calculate R=6 if V is constant with value V = 24. (b) Calculate R=6 if I is constant with value I = 4.

Current I (amperes), voltage V (volts), and resistance R (ohms) in a circuit are related by Ohms Law, I = V /R. (a) Calculate dI dR R=6 if V is constant with value V = 24. (b) Calculate dV dR R=6 if I is constant with value I = 4.

The tip speed ratio of a turbine (Figure 5) is the ratio R = T/W, where T is the speed of the tip of a blade and W is the speed of the wind. (Engineers have found empirically that a turbine with n blades extracts maximum power from the wind when R = 2/n.) Calculate dR/dt (t in minutes) if W = 35 km/h and W decreases at a rate of 4 km/h per minute, and the tip speed has constant value T = 150 km/h.

The curve y = 1/(x2 + 1) is called the witch of Agnesi (Figure 6) after the Italian mathematician Maria Agnesi (17181799), who wrote one of the first books on calculus. This strange name is the result of a mistranslation of the Italian word la versiera, meaning that which turns. Find equations of the tangent lines at x = 1. 3 2 1 1 2 3 1 x y FIGURE 6 The witch of Agnesi.

Let f (x) = g(x) = x. Show that (f/g) = f /g

Use the Product Rule to show that (f 2) = 2ff . 5

Show that (f 3) = 3f 2f .

Let f , g, h be differentiable functions. Show that (fgh) (x) is equal to f (x)g(x)h(x) + f (x)g (x)h(x) + f (x)g(x)h (x) Hint: Write fgh as f (gh). 58

Prove the Quotient Rule using the limit definition of the derivative.

Derivative of the Reciprocal Use the limit definition to prove d dx  1 f (x) = f (x) f 2(x) 7 Hint: Show that the difference quotient for 1/f (x) is equal to f (x) f (x + h) hf (x)f (x + h)

Prove the Quotient Rule using Eq. (7) and the Product Rule.

Use the limit definition of the derivative to prove the following special case of the Product Rule: d dx (xf (x)) = f (x) + xf (x)

Carry out Maria Agnesis proof of the Quotient Rule from her book on calculus, published in 1748: Assume that f , g, and h = f/g are differentiable. Compute the derivative of hg = f using the Product Rule, and solve for h .

The Power Rule Revisited If you are familiar with proof by induction, use induction to prove the Power Rule for all whole numbers n. Show that the Power Rule holds for n = 1; then write xn as x xn1 and use the Product Rule.

Exercises 64 and 65: A basic fact of algebra states that c is a root of a polynomial f if and only if f (x) = (x c)g(x) for some polynomial g. We say that c is a multiple root if f (x) = (x c)2h(x), where h is a polynomial. 64. Show that c is a multiple root of f if and only if c is a root of both f and f . 6

Use Exercise 64 to determine whether c = 1 is a multiple root: (a) x5 + 2x4 4x3 8x2 x + 2 (b) x4 + x3 5x2 3x + 2

Figure 7 is the graph of a polynomial with roots at A, B, and C. Which of these is a multiple root? Explain your reasoning using Exercise 64. x y A B C FIGURE 7

According to Eq. (8) in Section 3.2, d dx bx = m(b) bx. Use t