# Math 1151

Math 1151 Midterm 3 Makeup October 20, 2020 Name: OSU name.#: Page 1 of 6 Lecturer: Recitation Instructor: Recitation Time: Instructions: Show all relevant supporting work to receive full credit in Problems 2, 3, and 4. Incorrect answers with substantially correct work may receive partial credit. Unsubstantiated answers will receive no credit. Using a table of values or L’Hôpitals Rule are not considered valid support for computing limits. Using graphs not given in this assignment are not considered valid support for computing limits. Give exact answers unless instructed to do otherwise. Intervals should be expressed simplified in interval notation. If a requested value does not exist, write “DNE”. Write that limits “DNE” only if they do not exist and are neither ∞ nor −∞. You are expected to evaluate any trigonometric functions at standard angles. Calculators are permitted except those that have symbolic algebra or calculus capabilities. You are allowed to use your notes on this exam, as well as any materials in the Carmen course, but no other websites. All work on this exam must be your own. You are not allowed to post these questions online or ask for help from other people, or online sources. Your solutions are to be uploaded to Gradescope as a single pdf file. Failure to follow the Gradescope formatting guildelines will result in substantial penalties. SIGN YOUR NAME: Question Points 1 12 2 14 3 22 4 12 Total: 60 Score Math 1151 Midterm 3 Makeup Autumn 2020 1) The entire graph of a one-to-one function f is given in the figure below. Let f −1 be the inverse of f . 4 3 2 1 −4 −3 −2 −1 1 2 3 4 5 −1 −2 −3 −4 a) (2 points) f −1 (−3) = b) (2 points) d −1 f (x ) dx c) (2 points) d f dx 1 2x + = 2 x =2 = x =−3 d) Suppose f is the derivative of a function g , so f (x ) = g 0 (x ), with g continuous on the interval (−5, 4). i) (3 points) On which interval(s) is the function g increasing? I NT E RVAL(s ) : ii) (3 points) On which interval(s) is the function g concave down? I NT E RVAL(s ) : Math 1151 Midterm 3 Makeup Autumn 2020 2) Let f be a differentiable function on the interval (−2, 4) with values given in the table below. x −1 0 1 2 3 a) (6 points) Find the value of: f (x ) f 0 (x ) 7 −2 3 7 2 3 5 4 1 1 d f (1 + 3x ) . dx x =0 VALUE : b) (8 points) Find the value of: d sin( π2 x ) . (Hint: Use logarithmic differentiation.) f (x ) dx x =0 VALUE : Math 1151 Midterm 3 Makeup Autumn 2020 3) A cylinder is changing size in a such a way that the volume remains constant 18π cubic meters. Denote the radius of the cylinder by r , and the length of the cylinder by ` (both in meters), and the area of the circle on the side of the cylinder by dA A. At the instant ` = 2 meters, the length is shrinking at a rate of 2 meters/hour. Find the value of . dt `=3 Solve the problem by performing the following steps. a) (2 points) Label the diagram with the lengths r and `. b) (4 points) Find a formula that relates the length ` with the radius r . F ORMULA : c) (4 points) Find a formula that relates A and `, which does not involve r . F ORMULA : Math 1151 Midterm 3 Makeup d) (7 points) Find the value of dA dt Autumn 2020 . `=2 dA dt = `=2 e) (5 points) Write a sentence to explain what the rate found in (d) means about the cylinder. (Don’t forget UNITS.) Math 1151 Midterm 3 Makeup 4) Consider the curve given by: cos(xy ) = x (y − π) − 1 a) (7 points) Find y 0 = dy . dx dy = dx b) (5 points) Find an equation of the line tangent to the curve at the point (1, π). E QUAT I ON : Autumn 2020 Math 1151 Midterm 3 Makeup October 20, 2020 Name: OSU name.#: Page 1 of 6 Lecturer: Recitation Instructor: Recitation Time: Instructions: Show all relevant supporting work to receive full credit in Problems 2, 3, and 4. Incorrect answers with substantially correct work may receive partial credit. Unsubstantiated answers will receive no credit. Using a table of values or L’Hôpitals Rule are not considered valid support for computing limits. Using graphs not given in this assignment are not considered valid support for computing limits. Give exact answers unless instructed to do otherwise. Intervals should be expressed simplified in interval notation. If a requested value does not exist, write “DNE”. Write that limits “DNE” only if they do not exist and are neither ∞ nor −∞. You are expected to evaluate any trigonometric functions at standard angles. Calculators are permitted except those that have symbolic algebra or calculus capabilities. You are allowed to use your notes on this exam, as well as any materials in the Carmen course, but no other websites. All work on this exam must be your own. You are not allowed to post these questions online or ask for help from other people, or online sources. Your solutions are to be uploaded to Gradescope as a single pdf file. Failure to follow the Gradescope formatting guildelines will result in substantial penalties. SIGN YOUR NAME: Question Points 1 12 2 14 3 22 4 12 Total: 60 Score Math 1151 Midterm 3 Makeup Autumn 2020 1) The entire graph of a one-to-one function f is given in the figure below. Let f −1 be the inverse of f . 4 3 2 1 −4 −3 −2 −1 1 2 3 4 5 −1 −2 −3 −4 a) (2 points) f −1 (−3) = b) (2 points) d −1 f (x ) dx c) (2 points) d f dx 1 2x + = 2 x =2 = x =−3 d) Suppose f is the derivative of a function g , so f (x ) = g 0 (x ), with g continuous on the interval (−5, 4). i) (3 points) On which interval(s) is the function g increasing? I NT E RVAL(s ) : ii) (3 points) On which interval(s) is the function g concave down? I NT E RVAL(s ) : Math 1151 Midterm 3 Makeup Autumn 2020 2) Let f be a differentiable function on the interval (−2, 4) with values given in the table below. x −1 0 1 2 3 a) (6 points) Find the value of: f (x ) f 0 (x ) 7 −2 3 7 2 3 5 4 1 1 d f (1 + 3x ) . dx x =0 VALUE : b) (8 points) Find the value of: d sin( π2 x ) . (Hint: Use logarithmic differentiation.) f (x ) dx x =0 VALUE : Math 1151 Midterm 3 Makeup Autumn 2020 3) A cylinder is changing size in a such a way that the volume remains constant 18π cubic meters. Denote the radius of the cylinder by r , and the length of the cylinder by ` (both in meters), and the area of the circle on the side of the cylinder by dA A. At the instant ` = 2 meters, the length is shrinking at a rate of 2 meters/hour. Find the value of . dt `=3 Solve the problem by performing the following steps. a) (2 points) Label the diagram with the lengths r and `. b) (4 points) Find a formula that relates the length ` with the radius r . F ORMULA : c) (4 points) Find a formula that relates A and `, which does not involve r . F ORMULA : Math 1151 Midterm 3 Makeup d) (7 points) Find the value of dA dt Autumn 2020 . `=2 dA dt = `=2 e) (5 points) Write a sentence to explain what the rate found in (d) means about the cylinder. (Don’t forget UNITS.) Math 1151 Midterm 3 Makeup 4) Consider the curve given by: cos(xy ) = x (y − π) − 1 a) (7 points) Find y 0 = dy . dx dy = dx b) (5 points) Find an equation of the line tangent to the curve at the point (1, π). E QUAT I ON : Autumn 2020 Math 1151 Midterm 5 1. Suppose g is a continuous function on the interval [−2, 6] whose graph is given in the figure on the left below. y=g(x) -2 y=g(x) 3 3 2 2 1 1 1 -1 2 3 4 5 6 -2 -1 1 -1 2 3 4 5 -1 (a) (3 points) In the figure to the right above, sketch the rectangles corresponding to the LEFT Riemann sum of g on the interval [−2, 6] with n = 2. (b) (4 points) Find the value of the LEFT Riemann sum of g on the interval [−2, 6] with n = 2. VALUE: (c) (5 points) Find the value of the limit: lim g(x∗k )∆x on the interval [−2, 6]. n→∞ VALUE: Z 6 (3g(x) + 1) dx. (d) (5 points) Find the value of the definite integral −2 VALUE: 6 2. (a) (7 points) Find the general antiderivative of the function f (x) = 7(3x + 1)4 − √ 4 x+ x2 3 . +1 ANTIDERIVATIVE: (b) (7 points) Suppose a particle is traveling along a straight line with acceleration function a(t) = 5e3t + 3t. At time t = 0, the object has velocity v(0) = 2. Find the velocity function, v(t). v(t) = 3. (a) (6 points) Suppose f (x) = 3×4 − ex + 1 and g(x) = 4x − cos(2x), and set h(x) = f (x) . Find the g(x) form and calculate the limit: lim h(x). x→0 FORM: VALUE: (b) (8 points) According to the Compound Interest Formula, invested at an interest rate r, com $1 r n pounded n times per year will have a value of A(n) = 1 + dollars after 1 year (r is a positive n constant). Find the form and calculate the limit: lim A(n). n→∞ FORM: VALUE: 4. A right triangle in the first quadrant has been made with its lower left vertex at the origin, its lower right vertex at the point (x, 0), and its third vertex (x, y) on the graph of y = 4 − x2 with x and y measured in meters. 4 y 4 – x2 3 (x,y) 2 1 1 2 (a) (3 points) Find a formula for A(x) the AREA of the triangle whose lower right vertex is at the point (x, 0) (as a function of x). A(x) = (b) (4 points) What is the interval of interest for A(x)? Explain. x in: (c) (8 points) Find the value of x that gives the maximum area over all such triangles. (Don’t forget to JUSTIFY your answer.) x=

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