(b) (7 + 3x)5 (e) (4x 2 3)2 (x + 5)3 (h) 4 3x 2 x + 5 in .NET framework

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(c) (2x 3) 2 (f) (i) x+2 3 x 3 1 + x3
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15.11 Find the derivatives of the following functions: (a) 2x 3/4 (d) (7x 3 4x 2 + 2)1/4 (g)
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( f ) 8x 3/4 + 4x 1/4 x 1/3 (i) 4 (4 + x) x 1 ( j) (1 + x 3 )2/3 1 2x
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15.12 Find the slope-intercept equation of the tangent line to the graph of y = 15.13 Find the slope-intercept equation of the normal line to the curve y = 15.14 Let g(x) = x 2 4 and f (x) = x+2 . x 2
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x 2 + 16 at the point (3, 5).
(a) Find a formula for (g f )(x) and then compute (g f ) (x). (b) Show that the chain rule gives the same answer for (g f ) (x) as was found in part (a). 15.15 Find the absolute extrema of the following functions on the given intervals: x (a) f (x) = on [ 1, 1] (b) f (x) = (x 2)2 (x + 3)3 on [ 4, 3] 1 + x2 2 (c) f (x) = 5 4x on [ 1, 1] (d) f (x) = x x 2/3 on [0, 8] 3 1 (e) f (x) = x 2/5 x 7/5 on [ 1, 1] 9
CHAP. 15]
THE CHAIN RULE
15.16 Two towns, P and Q, are located 2 miles and 3 miles, respectively, from a railroad line, as shown in Fig. 15-3. What point R on the line should be chosen for a new station in order to minimize the sum of the distances from P and Q to the station, if the distance between A and B is 4 miles
Fig. 15-3
15.17 Assume that F and G are differentiable functions such that F (x) = G(x) and G (x) = F(x). If H(x) = (F(x))2 (G(x))2 , nd a formula for H (x). 15.18 If y = x 3 2 and z = 3x + 5, then y can be considered a function of z. Express dy/dz in terms of x. 15.19 Let F be a differentiable function, and let G(x) = F (x). Express Dx (F(x 3 )) in terms of G and x. 15.20 If g(x) = x 1/5 (x 1)3/5 , nd the domain of g (x). 15.21 Let f be a differentiable odd function (Section 7.3). Find the relationship between f ( x) and f (x). 15.22 Let F and G be differentiable functions such that F (3) = 5 G (3) = 7 If H(x) = F(G(x)), nd H (3). 15.23 Let F(x) = 1 + 3x F (3) = 13 G (3) = 6 F (7) = 2 G (7) = 0
(a) Find the domain and the range of F. (b) Find the slope-intercept equation of the tangent line to the graph of F at x = 5. (c) Find the coordinates of the point(s) on the graph of F such that the normal line there is parallel to the line 4x + 3y = 1. 15.24 Find the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 1 if a side of the rectangle is on the diameter. 15.25 Prove Theorem 15.1: If f is continuous at a and g is continuous at f (a), prove that g f is continuous at a. [Hint: For arbitrary > 0, let 1 > 0 be such that |g(u) g( f (a))| < whenever |u f (a)| < 1 . Then choose > 0 such that | f (x) f (a)| < 1 whenever |x a| < .] 15.26 Prove that x n/k is differentiable. [Hint: It is enough to show that f (x) = x 1/k (k > 1) is differentiable.] Proceed as follows: (1) By direct multiplication, establish that ak bk = (a b)(ak 1 + ak 2 b + + abk 2 + bk 1 ) 1 a b = k 1 k bk k 2 b + + abk 2 + bk 1 a a +a
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