MIDTERM 1. Name-Surname: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts Total. Overall 115 points.

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1 Name-Surname: Student No: Grade: 15 pts 20 pts 15 pts 10 pts 10 pts 10 pts 15 pts 20 pts 115 pts Total Overall 115 points. Do as much as you can. Write your answers to all of the questions. Grades greater than or equal to 100 will be 100. MAT 1025 CALCULUS I Dokuz Eylül Üniversitesi Fen-Edebiyat Fakültesi Matematik Bölümü Instructor: Engin Mermut MIDTERM 1 1. Find the following limits: x2 + 2 (a) lim x 2x

2 (b) lim x 0 + xx (c) ( 1 lim x 1 + x 1 1 ) ln x

3 2. Consider the function ( ) 1 x f(x) = 2 sin, if x 0; x 0, if x = 0. Explain and prove your answers to the following questions for the function f: (a) Is f a continuous function? (b) Is f differentiable at 0? If so, what is f (0)?

4 (c) What is f (x) for all x 0? (d) Is the function f continuous?

5 sin h 3. (a) Using only the limit fact lim h 0 h = 1 (and some trigonometric identities), prove that cos h 1 lim h 0 h = 0. (b) Using the limits in part (a) (and some trigonometric identities), find d (cos x) dx directly using the definition of derivative.

6 (c) The cosine function cos x is one-to-one on the interval [0, π] and so we can consider its inverse function cos 1 : [ 1, 1] [0, π]. Find d dy (cos 1 y) using the rule for the derivative of the inverse function: (f 1 ) (y) = 1 f (f 1 (y)).

7 4. (a) Let f(x) be a function defined on the unbounded interval [1, ) such that f is continuous on the interval [1, ) and differentiable on the open interval (1, ). Furthermore, assume that there exists a real number M > 0 such that f (x) M for all real numbers x > 1. Prove that for all real numbers a, b such that 1 a < b, f(b) f(a) M(b a). (b) For the function f(x) = x, using part (a), show that for all real numbers a, b such that 1 a < b, b a b a 2.

8 5. Suppose that for all x in an open interval containing π/4, the equation x sin 2y = y cos 2x has a unique solution for y in R which we denote by y = f(x). Further assume that y = f(x) is a differentiable function of x. Thus we have a differentiable function y = f(x) that has been implicitly defined for all x in an open interval containing π/4 such that (a) Find y = dy dx in terms of x and y. x sin 2y = y cos 2x. (b) Find the equation of the tangent line at the point (π/4, π/2) on the curve defined by the equation x sin 2y = y cos 2x.

9 6. The 8 meters wall shown in the figure stands 27 meters from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

10 7. Consider the hyperbolic tangent function tanh x = tanh x = sinh x cosh x = (ex e x )/2 (e x + e x )/2 = ex e x e x + e x. (a) Prove that d dx (tanh x) = sech2 x = fact that d dx (ex ) = e x (and chain rule of course). 1 cosh 2 x = 4 for all x R just by using the (e x + e x ) 2 (b) Prove that the inverse of the function tanh is given by tanh 1 x = 1 ( ) 1 + x 2 ln, x < 1. 1 x

11 (c) Find the derivative of the inverse hyperbolic tangent function tanh 1 x in two ways: i. By using the formula for tanh 1 x found in part (b); ii. By using the rule for the derivative of the inverse function: (f 1 ) (y) = 1 f (f 1 (y)).

12 8. (a) Draw the graph of the function by following the graphing strategy. f(x) = tanh x = sinh x cosh x = (ex e x )/2 (e x + e x )/2 = ex e x e x + e x (b) After you have drawn the graph of the function f, draw the graph of its inverse function f 1 = tanh 1. GRAPHING STRATEGY: (a) Identify the domain of f and any symmetries the curve may have. (b) Find f and f. (c) Find any critical points of f and identify the functions s behavior at each one. (d) Find where the curve is increasing and where it is decreasing. (e) Find the points of inflection, if any occur, and determine the concavity of the curve. (f) Identify any asypmtotes (horizontal asypmtotes and vertical asypmtotes). (g) Plot key points, such as the intercepts and the points founds in the steps (c), (d) and (e), and plot the asypmtotes if any. (h) Gathering all you have found in the previous steps, sketch the curve.