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Concept explainers
Let
(a) Find the equation of the secant fine through the points
(b) Show that there is only one point
(c) Find the equation of the tangent line to the graph of
(d) Use a graphing utility to generate the secant line in part (a) and the tangent line in part (c) in the same
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Chapter 4 Solutions
EBK CALCULUS EARLY TRANSCENDENTALS SING
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Calculus: Single And Multivariable
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Calculus: Early Transcendentals (3rd Edition)
Glencoe Math Accelerated, Student Edition
- Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.arrow_forward3. Let f(x) = 2x - x². (a) Find the secant line to the graph of f passing through the points (0,0) and (1,1). 1 (b) Find the tangent line to the graph of f at x = 2 (c) Sketch the graphs of f(x), the secant line, and tangent line on the same set of axes. -4 -3 -1 Y 5 2 4 -2 -4 6 2 3 A 5 xarrow_forwardThe curve of a function f (x) is given in the figure below and you are asked to draw on it a) Draw a tangent to the figure for x = 9 and explain in a few words what this line tells us about the curve of the function f (x). b) When interpreting f'(9), it is said that the result gives an approximate change of the function when x is increased by 1. Why is the relation approximately? Explain and draw on the picture to explain the deviation in question. c) Draw two secant lines between x = 9 and x = 6.5 otherwise on the one hand and x = 9 and x = 7 on the other. What is the usefulness of these profiles when discussing the instantaneous rate of change (e. Instantaneous Rate of Change) of the function in x = 9?arrow_forward
- Find the equation of the tangent line to the curve at the given point using implicit differentiation. Cardioid: (x2 + y2+ y)² = x² + y² at (1, 0) y The xy-coordinate plane is given. The curve starts at the point (1, 0), goes up and left becoming less steep, changes direction at the approximate point (0.4, 0.2), goes down and left becoming more steep, passes through the origin, sharply changes direction at the origin, goes up and left becoming less, steep, changes direction at the goes down and left becoming more steep, crosses the x-axis at the point (-1, 0), changes direction at the approximate point (-1.3, -0.7), goes down and right becoming less steep, changes direction at the approximate point (0, -2), goes up and right becoming more steep, changes direction at the approximate point (1.3, -0.7), goes up and left becoming less steep, and stops at the point (1, 0). oproximate point (-0.4, 0.2),arrow_forwardLet f be a differentiable function that satisfies Then f(x+y) -f(x) = 6 xy + 3y2 for any real numbers x,y . the derivative of f at x=-6, that is f(-6)= and the equation of the tangent line to f(x) passing through the point (-6,108) isarrow_forwardLet f be a function defined on the closed interval -4arrow_forwardThe second derivative of a function is given by f"(X) = xcosx. How many points of inflection does f have on the interval (-21, TT)? O O 4 06arrow_forward1. (#11 Exam 1 Review) Find the equation of the line tangent to the graph of the function: f (x) = x-2x² at x = 1. You must use the Definition of the Derivative to obtain the derivative. 2. (#9 Exam 1 Review) Sketch the graph of the function g for which: g(0) = g(2) = g(4) = 0, g' (1) = g(3) = 0,g' (2) = -1, lim g(x) = ∞, and lim g(x) = -∞ x 5 x-1+ 3. State the three-part Definition of Continuity. 4. (#13 Exam 1 Review) The graph of f is given. State, with mathematically correct reasons, the values for which f is not differentiable. -2 VA 0 2 4 X (Description: Graph with three components. First component is horizontal line in the 2nd quadrant to x = -2. Sharp turn upwards in downward arc into the 1st quadrant and ends with open dot where x = 1. Second component is solid dot at x = 1 vertically below the open dot. Third component begins with open dot at x = 1 below the solid dot. The graph falls in downward arc shape into the 4th quadrant, takes a sharp turn at x = 3 and continues as…arrow_forward17. Sketch the ioiicwing curve by cing seconá derivative: 1) y= 1+x 2) y=-x(x-7) 3) y (x+ 2) (x-3) 4) y=x(5-x) (ans.: max.(1,0.5); min.(-1,-0.5)) (ans.: max.(7,0); min.(2.3,-50.8)) (ans.: max.(-2,0); min.(1.3,-18.5)) (uns. mux.(3.5,18.5);ra0,0)) 18. What is the smallest perimeter possible for a rectangle of area 16 in.2 ? (ans.: 16) 19. Find the area of the largest rectangle with lower base on the x- axis and upper vertices on the parabola y 12-x. (ans.:32) 20) A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m. of fence at your disposal. What is the largest area you can enclose ? (ans.:80000) 21) Show that the rectangle that has maximum area for a given perimeter is a square. 22) A wire of length L is available for makıng a circie and a square. How should the wire be divided between the two shapes to maximize the sum of the enclosed areas? (ans.: all bent into a circle) 23) A closed container is made from a…arrow_forwardFind the indicated quantities for f(x) = 2x. (A) The slope of the secant line through the points (2,f(2)) and (2 + h,f(2 + h)), h 0 (B) The slope of the graph at (2,f(2)) (C) The equation of the tangent line at (2,f(2)) (A) The slope of the secant line through the points (2,f(2)) and (2 + h,f(2 + h)), h#0, is (B) The slope of the graph at (2,f(2)) is (Type an integer or a simplified fraction.) (C) The equation of the tangent line at (2,f(2)) is y =arrow_forwardFind and sketch the domain of the (real-valued) function f (x, y) = sin(/x² – y). - Find its range explaining your answer.arrow_forwardarrow_back_iosarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
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