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In Class Exam 2 Review Math 131 1. What is the limit definition of the derivative with respect to x as a function? Write the limit definition for the derivative of f ( x ) e x x 2 . 2. Find the equation of the tangent line to f(x) at (a, f(a)). a) f (x) 4x b) f (x) e x 3 4 2x 9 ln x a 16 a 1 3. Determine any and all x-values where the function is not differentiable. 2 5 a) f (x) x b) f (x) (x c) |x | 2 f (x) x 6 ( x 3 ) 12 2 9) 2 5 x 1 1 x 3 3 x In part c, what change could be made to make f continuous at x=3? Then would it be differentiable? 2 f (u ) u 2 4. h ( x ) f ( g ( x )) g(x) at x=1. Find the tangent line to h(x) at (1, h(1)). y x 4 is the tangent line to 5. Find f '(x) : a) x f (x) e b) x f (x) 2x 2 1 4x 7 x c) (x ln 2 3 4) ( e x 4 1 d) f ( x ) cos x e) f ( x ) sin 2 x x 2 f ) f ( x ) tan( x ) g) f ( x ) sec 5 (3 x ) 6. If f(t) is distance and t is time and f(t) is increasing at an increasing rate then v(t) is _______ and a(t) is _______________. When v(t) has a maximum, a(t) is ______________ and the graph of f(t) has ____________. If g '(x) is negative and is becoming more and more negative then g is __________and _________________. 7. Shown is the derivative of f(x) . Determine the intervals where f(x) is a) increasing, decreasing b) concave up, concave down c) Where does f (x) have a local max? a local min? d) Where does f( x) have inflection points? f '(x) 8. Use differentials or a linear approximation to approximate a) b) 3 3 66 63 9. An object travels in a straight line, east and west relative to its starting point.. Its velocity at t seconds is 3 2 v ( t ) t 8 t 9 t 18 a) When is its eastward velocity the greatest? b) When is its westward velocity the greatest? c) For 0<t<6, when is the objects position the farthest east of start? 10. Use the limit definition of the derivative to find d 3x 2 dx i) What is the slope of the secant line from (x, f(x)) to (a, f(a))? ii) Simplify until you can cancel x – a from the denominator. iii) Find the limit as x approaches a. 4 .