Exercise Sheet 2: Foundations of Differential Calculus

Transcription

1 D-ERDW, D-HEST, D-USYS Mathematics I Fall 2015 Dr. Ana Cannas Exercise Sheet 2: Foundations of Differential Calculus 1. Let the function f(x) be x 2 1, 1 x < 0, 2x, 0 < x < 1, f(x) = 1, x = 1, 2x + 4, 1 < x < 2, 0, 2 < x < 3, which is depicted in the following image. a) i. Does f( 1) exist? ii. Does lim x 1 + f(x) exist? iii. Is lim x 1 + f(x) = f( 1)? iv. Is f continous at the point x = 1? b) i. Does f(1) exist? ii. Does lim x 1 f(x) exist? iii. Is lim x 1 f(x) = f(1)? iv. Is f continous at the point x = 1? c) i. Is f defined at the point x = 2? (Check the definition of f.) ii. Is f continous at the point x = 2? 1

2 d) For what values of x is f(x) continous? e) What value has to be assigned to f(2)in order to make f(x) continous at the point x = 2? f) What value does f(1) need to be changed to, in order to eliminate the discontinouity? 2. Let f(x) be defined as f(x) = { 0, x 0 sin 1 x, x > 0. a) Does the limit lim x 0 + f(x) exist? If yes, what value is it? If no, why not? b) Does the limit lim x 0 f(x) exist? If yes, what value is it? If no, why not? c) Does the limit lim x 0 f(x) exist? If yes, what value is it? If no, why not? d) For what values of x is f continous? 3. Calculate the following derivatives of the functions f(x) = (x 1)(x 2 + 3x 5) t 1 g(t) = t + 1 a) f (x) by applaying the product rule. h(t) = 1 + t 4t2 t b) f (x) by expanding the product which leads to a easy differentiatable sum of terms. c) All orders of derivative of f(x). 2

3 d) g (1) e) h (1) f) (gh) (1) by applaying the product rule. 4. Exponential functions of the form y 0 e at describe exponential growth if a > 0 and exponential decayl forr a < 0. The constant y 0 kan be understood as initial value and the variable t is a number. For the following tasks you are allowed to use a calculator. a) A model for the growth of a investement of y 0 Chf. assuming a annual rate of interest of r% is given by y(t) = y 0 e r 100 t, where t stands for the number of years since the start of the investment. 1 If we invest 100Chf. today with a rate of interest of 5% how much can we expect after 4 years? b) The radioaktive decay of a radioaktive sumstance (for example uranium, radium or carbon-14) can be described as where N (t) = N 0 e λt, N (t) is the number of atomic nuclei of the radioaktive substance present at time t. N 0 = N (0) isis the number the number of atomic nuclei initially present. Verify that λ 0.693, τ where τ is the socalled half-life, this is the period in which excactly half of all atomic nuclei have decayed N 0 have decayed: N (τ) = 1 2 N 0. For λ = ,where t is mesured in years (this corresponds to experimental determination of the rate of decay of carbon-14), make a prediction regarding the percentage of the initial substancel N 0 still present after 866 years. 1 The first approximation of the numer e appeared in a work of the swiss mathematician Jakob Bernoulli ( ) exactly in the context of calcultaions of rete of interest. 3

4 c) Verify that the functions y(t) and N (t) satisfy the equations ẏ = r 100 y and N = λn where f means the derivative of f with respect to the time. 5. A function which reverses or inverts hte assignment rule of a other function f is called inverse function of f. Inverse functions play a big role in applications in natural sciences. The most importent ones are the exponentil and logarithmic functions. The natural logarithm or ln-function is defined by ln : R + R y = e x ln y = x. This means that lny is the exponent to which power e has to be raised in order to obtain y. a) What is the value of ln 1? What is the value of ln (e π )? b) Sketch the graph of ln x. c) Determine d dy ln y using the rule for the derativative of a inverse. d) Determine the derivative of e) What is the inverse function of e 3x? x ln x x f) Determine the derivative at e 3 of the inverse function e 3x using the rule for the derativative of a inverse. 6. the general exponential functionsare the functions of the form where the base a is positive, a 1 andthe exponent is variable. a) Choose k as a real number such that Hint: Logarithm. a x, a x = e kx. 4

5 b) Calculate the derivative of a x unsing the chain rule and the result obtained in part a). c) Show that a x is strictly monotonic increasing if a > 1 and strictly monotonic decreasing if 0 < a < 1. d) Sketch the graph of a x. You have to distinguish the cases a > 1 and 0 < a < 1. e) We are looking at the fucntion g(a) = a X, for a > 0 with X a real positive constant. Calculate the derivative of a X with respect to a. 7. The logarithmic function to the base a, log a x, is the inverse function of a x : y = a x log a y = x. Where log e y = ln y is the usual (natural) Logarithm. a) What is d dy log a y? Hint: Formula for the derivative of a inverse function. b) Calculate the following numbers (without calculator): c) Using the power rules log , log , log 2 64, log a x a x 0 = a x+x 0 1 a = x a x (a x ) k = a x k und derive the corresponding rules for logarithmic functions log a (y y 0 = ( ) 1 log a = y log a (y k ) = with respect to log a y and log a y 0. d) Sum up the following expression: 2 log a (3x) + log a (2x) 1 2 log a(9x 2 ). 5

6 e) Solve the equation e (ln 0.2)t = f) Careful: a expressions such as 2 34 can be ambiguous! Which number is bigger (2 3 ) 4 oder 2 (34)? Without using a calculator. g) Apply the logarithm on both sides of the equation y = 1 10 x 2 3 in order to rewrite them with respect to the new variables X = log 10 x and Y = log 10 y. The solutions are: 1. a) yes, yes, yes, yes b) yes, yes, no, no c) no, no d) for x in [ 1, 0), (0, 1),(1, 2) and (2, 3) e) 0 f) 2 2. a) no b) yes, 0 c) no d) for x in (, 0) and (0, + ). 3. a) f (x) = 8 + 4x + 3x 2 b) f (x) = 8 + 4x + 3x 2 c) f (x) = 4 + 6x, f (x) = 6, f (x) = 0 d) 1 4 e) 5 f) a) 122,15 CHF b) After 866 years there are approximately 90, 1 percent of the initial substance remaining. 5. a) ln(1) = 0, ln(e π ) = π 6

7 c) 1 y d) ln(x) e) ln(y) 3 f) 1 3e 3 6. a) k = ln(a) b) ln(a)a x. e) Xa X 1 7. a) 1 y ln(a) b) 3, 0.5, 6, 1 c) i. log a (yy 0 ) = log a y + log a y 0 ii. log a (y 1 ) = log a y iii. log a y k = k log a y d) log a (6x 2 ) e) t = 2 f) (2 3 ) 4 < 2 (34 ) g) Y = X 7

8 MC-Sheet 2 1. Which of the following formulas is not a valid calculation rule for all x, y, z > 1? a) (x y ) z = x (yz). x y b) x = y z xz. c) log x x yz = yz. d) log x y z = z log x y. 2. The inverse function g(y) of f(x) = 1 1+2e x on the interval (0, 1) is: a) g(y) = ln ( 1 + 2e y). ( ) 1 y b) g(y) = ln. 2y c) g(y) = 1 + 2e y. d) g(y) = e y. 3. In the following image the red line is tangential to the blue curve in point P. The curve corresponds to a function f : R R. What is the value of the derivative f at point 1? f x P 1,2 3 a) 2 b) 1 2 c) 2 3 d) 2 8

9 f x g x h x 3 3 x x 3 3 x 4. The following three pictures show the graphs of three real valued functions with real valued variables f(x), g(x) and h(x) where one is the derivative of the other. Which statement is true? a) f = g. b) g = f. c) f = h. d) h = g. 5. What is the slope of the tangent at x = π2 4 a) 1. b) π. c) 1 π. d) π2 2. of the graph of f(x) = cos x? 6. Which of the following functions is strictly monotonic increasing on the interval ] 1, 1[? a) x x 2 b) x x + x c) x e x d) x arccos x 7. What is the derivative of f(x) = 2 x2 +1? a) 2x 2 x2. 9

10 b) (x 2 + 1) 2 x2. c) (ln 2)2 x2 +1. d) 4x(ln 2)2 x2. 8. Which of the following statements is wrong? The derivative of the function a) x(t) = sin ( e 2t) ist ẋ(t) = 2e 2t cos ( e 2t). b) x(t) = 1 t + t ln t, t > 0, ist ẋ(t) = 2 + ln t + t. 2 t2 c) x(t) = e ln t+t2, t > 0, ist ẋ(t) = ( 1 + 2t 2) e t2. ( ist ẋ(t) = 2t sin(t 2 ) 1 + d) x(t) = sin2 (t 2 ) cos(t 2 ) 9. The equation 1 cos 2 (t 2 ) y 2 = x 2 sin(xy) + 1 defines y as differentiable function of x in the surroundings of (x, y) = (0, 1). What is the derivative dy of this function? dx ). a) b) c) d) 2y + x cos(xy) 2x y cos(xy). 2y + x cos(xy) 2x + y cos(xy). 2x y cos(xy) 2y + x cos(xy). 2x + y cos(xy) 2y + x cos(xy). 10. Let f(x) = π + 2 arctan(x) and g(x) the inverse function of f(x). Which of the following statements is wrong? a) f(x) is defined and differentiable for all real values x. b) The range of f(x) is (0, 2π). c) g(x) is defined and differentiable for all real values x. ( d) g(x) = cot π x ) 2 10