By Dovermann K.H.

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11. Use l(x) = − sin(x0 )(x − x0 ) + cos x0 as the proposed tangent line to the graph of cos x at (x0 , cos x0 ). 21)): cos(x0 + h) = cos x0 cos h − sin x0 sin h. CHAPTER 2. 12. Let a and c be constants. The function f (x) = ceax is differentiable at all x, and f (x) = aceax . Furthermore, functions of the form f (x) = ceax are the only functions which satisfy the equation f (x) = af (x). Exercise 38. Show that the exponential function exp x = ex is its own derivative. At this point we are not in the position to prove either statement in the theorem.

SOME BACKGROUND MATERIAL 26 Furthermore, loga (u) = loga (v) implies that u = aloga (u) = aloga (v) = v. This verifies the remaining claim in the theorem. 13 on page 22 we have the laws of logarithms. 11. The other parts are assigned as exercises below. 19 (Laws of Logarithms). For any positive real number a = 1, for all positive real numbers x and y, and any real number z loga (1) = 0 loga (a) = 1 loga (xy) = loga (x) + loga (y) loga (x/y) = loga (x) − loga (y) loga (xz ) = z loga (x) Because the exponential and logarithm functions are inverses of each other, their rules are equivalent.

That means that the tangent line has the formula l(x) = e(x − 1) + e = ex. Our goal is to find a line which is close to the graph, near a given point. So let us check how close l(x) is to ex if x is close to 1. 1 you find the values of ex and l(x) for various values of x. You see that ex − l(x) is small, particularly for x close to 1. Let us compare ex − l(x) and x − 1 by taking their ratio (ex − l(x))/(x− 1). As you see in the second last column of the table, even this quantity is small for x near 1.