Question 1 In this question we will estimate the value of tan (0.34x) using the linearization of f(I) = tan(NZ) at the nearby point : = 1/3. a) Compute the derivative of f at 1/3. f' (1/3) = FORMATTING: 1 is written pi, va is written sqrt(a). b) Write down the linearization L(x) of tan(xx) at x = 1/3. L(I) = FORMATTING: Give the exact value of all coefficients. c) Since 0.34 is near 1/3, we use the linear approximation of tan(x) found in (b) to estimate the value of tan(x) at x = 0.34. This gives tan(0.347) ~ Number FORMATTING: Round your answer to at least six decimal places. (You should check with your calculator and judge if this value is a good estimate.)- Question 2 In this question, we will estimate the value of (6/5) 1/4using a linearization of f(x) = (1 + 6x)1/4 a) Find f (0) = Number b) Find the linearization L(x) of f(x) at the point : = 0. L(I) = FORMATTING: Your answer must be a function of I. c) Now work out for what value of I we have f() = (6/5) 1/4 Answer: I = Number d) Since your answer in (c) is close to 0, we may use our linearization in (b) to estimate (6/5)1/4 Answer = Number You may verify with your calculator that this answer is close to the true value.- Question 3 Given that L(x) = 2+31 is the linearization of a mystery function f() at x = 4, what are the values of f(4) and f (4)? (a) f(4) = Number (b) f ( 4 ) = Number- Question 4 813 +11x-6 We will find the general form of an antiderivative F(x) of the function f(x) = on the domain r * 0. a) Simplify: express f(c) as a sum of terms of the form Bar, where o and B are some real numbers. Answer: f (I) = b) Use your answer in (a) to find the general antiderivative of f. Answer: F(x) = DE +c FORMATTING: Do not include the constant of integration because we did it for you.\fQuestion 6 The rate of acceleration due to gravity on Mars is approximately 3.8 m/s2. Air resistance is negligible. You stand at the top of a cliff and drop a wrench. Using "up" as the positive direction, and with t = 0 corresponding to when you dropped the wrench, answer the following questions, paying attention to signs. (a) What is the velocity function of the wrench as it falls? u(t) = (b) What is the displacement function of the wrench, if your altitude when you stand at the top of the cliff is measured as 1,800 m by a GPS? s (t) = Jim (altitude) (c) If the wrench hits the ground at the bottom of the cliff with a speed of 22.8 m/s, what is the height of the cliff (that is, the difference in altitude between the top of the cliff and the ground)? h =- Question 7 The general form of an antiderivative of the function f (x) = sec (x) (8 tan (x) - 3 sec (z)) is F () + C Do not include the integration constant in the answer.- Question 8 The sigma notation for the sum is a standard notation in mathematics used to concisely express sums with a great number of terms. See your textbook for more information. Associate the equivalent expressions: 1. 50 + 1 + 12 + 13 + ... + In 2. nf(=1) 3. f(1 1 ) + f (12) + ... + f (In) 4. f(=0 ) + f ( 1 1 ) + f ( 32 ) + ...+ f(In)