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: f(θ) = 4 sin θ − θ. Point: (0, 0 )
> Find an equation of the tangent line to the graph of the function f through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation (x0, y0) = (5, 0)
> Find an equation of the tangent line to the graph of the function f through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the equation f(x) = √x (x0,y0) = ( - 4, 0)
> Show that the graph of the function f (x) = x5 + 3x3 + 5x
> Show that the graph of the function f (x) = 3x + sin x + 2 does not have a horizontal tangent line.
> Show that the graphs of the two equations have tangent lines that are perpendicular to each other at their point of intersection.
> Sketch the graphs of y = x2 and y = −x2 + 6x − 5, and sketch the two lines that are tangent to both graphs. Find equations of these lines.
> Find the derivative of the function.
> Use the graph of f to answer each question. To print an enlarged copy of the graph, go to MathGraphs.com. (a) Between which two consecutive points is the average rate of change of the function greatest? (b) Is the average rate of change of the functio
> Sketch the graph of a function f such that f′ > 0 for all x and the rate of change of the function is decreasing.
> The graphs of a function f and its derivative f′ are shown on the same set of coordinate axes. Label the graphs as f or f′ and write a short paragraph stating the criteria you used in making your selection. To print an
> The graphs of a function f and its derivative f′ are shown on the same set of coordinate axes. Label the graphs as f or f′ and write a short paragraph stating the criteria you used in making your selection. To print an
> g(x) = 3 f(x) − 1 The relationship between f and g is given. Explain the relationship between f′ and g′.
> The relationship between f and g is given. Explain the relationship between f′ and g′. g(x) = −5 f(x)
> The relationship between f and g is given. Explain the relationship between f′ and g′. g(x) = 2 f(x)
> g(x) = f(x) + 6 The relationship between f and g is given. Explain the relationship between f′ and g′.
> Find k such that the line is tangent to the graph of the function. Function: f(x) = k√x Line: y = x + 4
> Find k such that the line is tangent to the graph of the function. Function: / Line: /
> Find the derivative of the function.
> Function: f(x) = kx2 Line: = −2x + 3 Find k such that the line is tangent to the graph of the function.
> Function: f(x) = k − x2 Line: y = −6x + 1 Find k such that the line is tangent to the graph of the function.
> y = √3x + 2 cos x, 0 ≤ x < 2π Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x + sin x, 0 ≤ x < 2π Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x2 + 9 Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x3 + x Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> y = x4 − 2x2 + 3 Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> Find the derivative of the function.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> f(x) = 6√x + 5 cos x Find the derivative of the function.
> f(t) = t 2/3 − t 1/3 + 4 Find the derivative of the function.
> f(x) = √x − 6√ 3 x Find the derivative of the function.
> y = x2 (2x2 − 3x) Find the derivative of the function.
> y = x(x2 + 1) Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> Find the derivative of the function.
> f(x) = x3 − 2x + 3x−3 Find the derivative of the function.
> f(x) = x2 + 5 − 3x−2 Find the derivative of the function
> Function: g(t) = −2 cos t + 5 Point: (π, 7) 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.
> 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
> 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). Dra
> 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
