Question: The figure below shows the topographic map
The figure below shows the topographic map carried by a group of hikers. The hikers are in a wooded area on top of the hill shown on the map, and they decide to follow the path of steepest descent (orthogonal trajectories to the contours on the map). Draw their routes if they start from point A and if they start from point B. Their goal is to reach the road along the top of the map. Which starting point should they use?
> f(x) = x(2x - 5)3 Find the derivative of the function.
> Function: f(x) = 2(x − 4)2 Point: (2, 8) Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results.
> Function: y = 2x4 − 3 Point: (1, -1) Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results.
> Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function: / Point:
> Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function: / Point: (4, 1)
> Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. Function: / Point: (2,2)
> Complete the table to find the derivative of the function. Original Function: / Rewrite: Differentiate: Simplify:
> Complete the table to find the derivative of the function. Original Function: / Rewrite: Differentiate: Simplify:
> Complete the table to find the derivative of the function. Original Function: Rewrite: Differentiate: Simplify:
> Complete the table to find the derivative of the function. Original Function: / Rewrite: Differentiate: Simplify:
> f(x) = x2 (x - 2)7 Find the derivative of the function.
> y = 7x4 + 2 sin x Use the rules of differentiation to find the derivative of the function.
> y = x2 – ½ cos. Use the rules of differentiation to find the derivative of the function.
> g(t) = π cos t Use the rules of differentiation to find the derivative of the function.
> Use the rules of differentiation to find the derivative of the function.
> y = 2x3 + 6x2 − 1 Use the rules of differentiation to find the derivative of the function.
> s(t) = t 3 + 5t 2 − 3t + 8 Use the rules of differentiation to find the derivative of the function.
> y = 4x − 3x3 Use the rules of differentiation to find the derivative of the function.
> g(x) = x2 + 4x3 Use the rules of differentiation to find the derivative of the function.
> y = t 2 − 3t + 1 Use the rules of differentiation to find the derivative of the function.
> f(t) = −3t 2 + 2t − 4 Use the rules of differentiation to find the derivative of the function.
> Find the derivative of the function.
> g(x) = 6x + 3 Use the rules of differentiation to find the derivative of the function.
> f(x) = x + 11 Use the rules of differentiation to find the derivative of the function.
> Use the rules of differentiation to find the derivative of the function.
> Use the rules of differentiation to find the derivative of the function. /
> Use the rules of differentiation to find the derivative of the function.
> Use the rules of differentiation to find the derivative of the function.
> y = x12 Use the rules of differentiation to find the derivative of the function.
> y = x7 Use the rules of differentiation to find the derivative of the function.
> f(x) = −9 Use the rules of differentiation to find the derivative of the function.
> y = 12 Use the rules of differentiation to find the derivative of the function.
> Find the derivative of the function.
> Use the graph to estimate the slope of the tangent line to y = xn at the point (1, 1). Verify your answer analytically. To print an enlarged copy of the graph, go to MathGraphs.com. (a) y = x−1/2 (b) y = x−1
> Use the graph to estimate the slope of the tangent line to y = xn at the point (1, 1). Verify your answer analytically. To print an enlarged copy of the graph, go to MathGraphs.com (a) y = x1/2 (b) y = x3
> Describe the difference between average velocity and velocity.
> What are the derivatives of the sine and cosine functions?
> Explain how to find the derivative of the function f(x) = cxn.
> What is the derivative of a constant function?
> (a)Find an equation of the normal line to the ellipse x2/32 + y2/8 = 1 at the point (4, 2). (b) Use a graphing utility to graph the ellipse and the normal line. (c) At what other point does the normal line intersect the ellipse?
> The graph shows the normal lines from the point (2, 0) to the graph of the parabola x = y2. How many normal lines are there from the point (x0, 0) to the graph of the parabola if (a) x0 = ¼, (b) x0 = ½, and (c) x0 = 1? (d)
> Find equations of both tangent lines to the graph of the ellipse x2/4 + y2/4 = 1 that pass through the point (4, 0) not on the graph.
> Find all points on the circle x2 + y2 = 100 where the slope is 3/4.
> Find the derivative of the function.
> What is the difference between the (Simple) Power Rule and the General Power Rule?
> Prove (Theorem 2.3) that for the case in which n is a rational number. (Hint: Write y = xp/q in the form yq = xp and differentiate implicitly. Assume that p and q are integers, where q > 0.)
> Let L be any tangent line to the curve Show that the sum of the x- and y- intercepts of L is c.
> Consider the equation x4 = 4(4x2 – y2) Use a graphing utility to graph the equation. Find and graph the four tangent lines to the curve for y = 3 Find the exact coordinates of the point of intersection of the two tangent lines in the first quadrant.
> Use the graph to answer the questions. Which is greater, the slope of the tangent line at x = -3 3 or the slope of the tangent line at x = -1? Estimate the point(s) where the graph has a vertical tangent line. Estimate the point(s) where the graph has
> Verify that the two families of curves are orthogonal, where C and K are real numbers. Use a graphing utility to graph the two families for two values of C and two values of K. x2 + y2 = C2 y = Kx
> xy = C, x2 – y2 = k Verify that the two families of curves are orthogonal, where C and K are real numbers. Use a graphing utility to graph the two families for two values of C and two values of K.
> x3 = 3(y - 1) x(3y - 29) = 3 Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each othe
> Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] x + y = 0 x = sin y
> Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] y2 = x3 2x2 + 3y2 = 5
> Find the derivative of the function.
> Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] 2x2 + y2 = 6 y2 = 4x
> 4x2 + y2 – 8x + 4y + 4 = 0 Find the points at which the graph of the equation has a vertical or horizontal tangent line.
> 25x2 + 16y2 + 200x – 160y + 400 = 0 Find the points at which the graph of the equation has a vertical or horizontal tangent line.
> Two circles of radius 4 are tangent to the graph of y2 = 4x at the point (1, 2). Find equations of these two circles.
> Show that the normal line at any point on the circle x2 + y2 = r2 passes through the origin.
> Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the circle, the tangent lines, and the normal lines. x2
> x2 + y2 = 25 (4, 3), (-3, 4) Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the circle, the tangent l
> Use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window.
> Use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window.
> 3xy − 4 cos x = −6 Find d2y/dx2 implicitly in terms of x and y.
> Find the derivative of the function.
> 7xy + sin x - 2 Find d2y/dx2 implicitly in terms of x and y.
> xy – 1 = 2x + y2 Find d2y/dx2 implicitly in terms of x and y.
> x2y – 2 = 5x + y Find d2y/dx2 implicitly in terms of x and y.
> x2y – 4x = 5 Find d2y/dx2 implicitly in terms of x and y.
> x2 + y2 = 4 Find d2y/dx2 implicitly in terms of x and y.
> Find dy/dx implicitly and find the largest interval of the form –a < y < a or 0 < y < a such that y is a differentiable function of x. Write dy/dx as a function of x. cos y = x
> Find dy/dx implicitly and find the largest interval of the form –a < y < a or 0 < y < a such that y is a differentiable function of x. Write dy/dx as a function of x. tan y = x
> Use implicit differentiation to find an equation of the Show that the equation of the tangent line to the ellipse
> Use implicit differentiation to find an equation of the Show that the equation of the tangent line to the ellipse
> Explain why the derivative of x2 + y2 + 2 = 1 does not mean anything.
> Find the derivative of the function.
> Write two different equations in implicit form that you can write in explicit form. Then write two different equations in implicit form that you cannot write in explicit form.
> Find an equation of the tangent line to the graph at the given point. Kappa curve
> Find an equation of the tangent line to the graph at the given point. Lemniscate
> Find an equation of the tangent line to the graph at the given point. Astroid
> Find an equation of the tangent line to the graph at the given point. Cruciform
> Find an equation of the tangent line to the graph at the given point. Circle
> Find an equation of the tangent line to the graph at the given point. Parabola
> Folium of Descartes: x3 + y3 - 6xy = 0 Find the slope of the tangent line to the graph at the given point.
> Find the slope of the tangent line to the graph at the given point. Bifolium: (x2 + y2)2 = 4x2y
> Find the slope of the tangent line to the graph at the given point. Cissoid: (4 - x)y2 = x3
> Find the derivative of the function.
> Find the slope of the tangent line to the graph at the given point. Witch of Agnesi: (x2 + 4)y = 8
> x cos y = 1, (2, π/3) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> tan(x + y) = x, (0, 0) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> x3 + y3 – 6xy – 1, (2, 3) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> (x + y)3 = x3 + y3, (-1, 1) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> 3x3y = 6, (1, 2) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> xy = 6, (-6, -1) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> (a) find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx implicitly
> Find the derivative of the function.
> (a) find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx implicitly
