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CALCULUS I YILDIZ TECHNICAL UNIVERSITY FATIH TEMIZ Differentiation Recall the definitions of limits and continuity. See the previous application lesson. 1. Determine the continuity of the function f (x) = 3|x − 2| . Classify the discontinuous points if there x2 (4 − x2 ) exist. 3x+4 2. Determine whether the function f (x) = x = 2 or not. e x−2 has limit at x = 2. Also determine if it is continuous at Derivative Function The derivative of the function f (x) with respect to the variable x is the function f 0 whose value at x is f (x + h) − f (x) f (z) − f (x) lim = lim z→x h→0 h z−x provided the limit exists. 3. By using the definition of derivative, find the derivative of the given functions below p (a) f (x) = x2 + 1 (b) f (x) = tan(2x) at x = π/6 (c) |f (x)| ≤ x2 at x = 0 2016 − 2017 CALCULUS I YILDIZ TECHNICAL UNIVERSITY FATIH TEMIZ (d) at x = 0 (√ x(1 − cos x) x ≥ 0 sin x x<0 3 f (x) = 4. Find the values of a and b in order to make the function given below differentiable at x = 0. ( ax + b x<0 f (x) = 2 sin x + 3 cos x x ≥ 0 5. Find the equation of the tangent to the curve 1 1 1 (a) y sin( ) + x cos( ) = −2x at the point P (0, ) y y π (b) x2 y + y 2 = 3x2 at the point (2, 2) (c) cos(x − y) = xex at the point P (0, π ) 2 2016 − 2017 CALCULUS I YILDIZ TECHNICAL UNIVERSITY FATIH TEMIZ Linearization If f is differentiable at x = a, then the approximating function L(x) = f (a) + f 0 (a)(x − a) is the linearization of f at a. The approximation f (x) ≈ L(x) of f by L is the standard linear approximation of f at a. 6. Find the approximate value of √ 3 28 by using differential calculus or linear approximation. 7. Let g : R → R be a differentiable function with g(2) = −4 and g 0 (x) = value of g(2.05) by using linear approximation. √ Derivatives of Trigonometric Functions d (sin x) = cos x dx d (tan x) = sec2 x dx d (cot x) = − csc2 x dx d (cos x) = − sin x dx d (sec x) = sec x tan x dx d (csc x) = − csc x cot x dx 8. Find the derivative of the functions w.r.t. x (a) (1 + ln x)x (b) x3 p + 3 (1 + x2 )3 q 3 x+ √ x 2016 − 2017 x2 + 5. Find the approximate CALCULUS I YILDIZ TECHNICAL UNIVERSITY FATIH TEMIZ (c) xy = cot(xy) (d) p 3 1 + cos(2x) (e) x2 y + xy 2 = 6. 9. Let f (x) and g(x) be two differentiable functions. Assume that f (g(x)) = x and f 0 (x) = 1 + (f (x))2 . 1 . Then show that g 0 (x) = 1 + x2 10. Determine whether the function f (x) = arcsin( is constant, find it. √ 1−x ) + 2 arctan( x) for x ≥ 0 is constant or not. If it 1+x 2016 − 2017