Calculus (MindTap Course List)
Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Chapter 2.1, Problem 25E

Sketch the graph of a function g that is continuous on its domain ( − 5 , 5 ) and where g ( 0 ) = 1 , g ′ ( 0 ) = 1 , g ′ ( − 2 ) = 0 , lim x → − 5 + g ( x ) = ∞ , and lim x → 5 − g ( x ) = 3 .

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Chapter 2 Solutions

Calculus (MindTap Course List)

Ch. 2.1 - A curve has equation y=f(x). a Write an expression...Ch. 2.1 - Graph the curve y=sinx in the viewing rectangles...Ch. 2.1 - a Find the slope of the tangent line to the...Ch. 2.1 - a Find the slope of the tangent line to the curve...Ch. 2.1 - Prob. 5ECh. 2.1 - Prob. 6ECh. 2.1 - Find an equation of the tangent line to the curve...Ch. 2.1 - Find an equation of the tangent line to the curve...Ch. 2.1 - a Find the slope of the tangent to the curve...Ch. 2.1 - a Find the slope of the tangent to the curve y=1/x...
Ch. 2.1 - a A particle starts by moving to the right along a...Ch. 2.1 - Shown are graphs of the position functions of two...Ch. 2.1 - If a ball is thrown into the air with a velocity...Ch. 2.1 - If a rock is thrown upward on the planet Mars with...Ch. 2.1 - Prob. 15ECh. 2.1 - The displacement in feet of a particle moving in a...Ch. 2.1 - Prob. 17ECh. 2.1 - The graph of a function f is shown. a Find the...Ch. 2.1 - Prob. 19ECh. 2.1 - Find an equation of the tangent line to the graph...Ch. 2.1 - If an equation of the tangent line to the curve...Ch. 2.1 - If the tangent line to y=f(x) at 4, 3 passes...Ch. 2.1 - Sketch the graph of a function f for which f(0)=0,...Ch. 2.1 - Sketch the graph of a function g for which...Ch. 2.1 - Sketch the graph of a function g that is...Ch. 2.1 - Prob. 26ECh. 2.1 - Prob. 27ECh. 2.1 - Prob. 28ECh. 2.1 - Prob. 29ECh. 2.1 - Prob. 30ECh. 2.1 - Prob. 31ECh. 2.1 - Prob. 32ECh. 2.1 - Find f(a). f(x)=2t+1t+3Ch. 2.1 - Prob. 34ECh. 2.1 - Find f(a). f(x)=12xCh. 2.1 - Prob. 36ECh. 2.1 - Each limit represents the derivative of some...Ch. 2.1 - Prob. 38ECh. 2.1 - Each limit represents the derivative of some...Ch. 2.1 - Prob. 40ECh. 2.1 - Prob. 41ECh. 2.1 - Prob. 42ECh. 2.1 - Prob. 43ECh. 2.1 - A particle moves along a straight line with...Ch. 2.1 - Prob. 45ECh. 2.1 - A roast turkey is taken from an oven when its...Ch. 2.1 - Researchers measured the average blood alcohol...Ch. 2.1 - Prob. 48ECh. 2.1 - Prob. 49ECh. 2.1 - The table shows values of the viral load V(t) in...Ch. 2.1 - The cost in dollars of producing x units of a...Ch. 2.1 - If a cylindrical tank holds 100, 000 gallons of...Ch. 2.1 - The cost of producing x ounces of gold from a new...Ch. 2.1 - The number of bacteria after t hours in a...Ch. 2.1 - Let H(t) be the daily cost in dollars to heat an...Ch. 2.1 - The quantity in pounds of a gourmet ground coffee...Ch. 2.1 - The quantity of oxygen that can dissolve in water...Ch. 2.1 - The graph shows the influence of the temperature T...Ch. 2.1 - Prob. 59ECh. 2.1 - Prob. 60ECh. 2.1 - Prob. 61ECh. 2.2 - Use the given graph to estimate the value of each...Ch. 2.2 - Use the given graph to estimate the value of each...Ch. 2.2 - Match the graph of each function in ab with the...Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Shown is the graph of the population function P(t)...Ch. 2.2 - A rechargeable battery is plugged into a charger....Ch. 2.2 - The graph from the US Department of Energy shows...Ch. 2.2 - The graph shows how the average age M of first...Ch. 2.2 - Make a careful sketch of the graph of the sine...Ch. 2.2 - Let f(x)=x2. a Estimate the values of...Ch. 2.2 - Let f(x)=x3. a Estimate the values of...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Prob. 28ECh. 2.2 - Prob. 29ECh. 2.2 - a Sketch the graph of f(x)=6x by starting with the...Ch. 2.2 - Prob. 31ECh. 2.2 - Prob. 32ECh. 2.2 - Prob. 33ECh. 2.2 - The table gives the number N(t), measured in...Ch. 2.2 - Prob. 35ECh. 2.2 - Prob. 36ECh. 2.2 - Let P represent the percentage of a citys...Ch. 2.2 - Prob. 38ECh. 2.2 - The graph of f is given. State, with reasons, the...Ch. 2.2 - The graph of f is given. State, with reasons, the...Ch. 2.2 - The graph of f is given. State, with reasons, the...Ch. 2.2 - The graph of f is given. State, with reasons, the...Ch. 2.2 - Prob. 43ECh. 2.2 - Prob. 44ECh. 2.2 - The graphs of a function f and its derivative f...Ch. 2.2 - The graphs of a function f and its derivative f...Ch. 2.2 - The figure shows the graphs of f,f,andf. Identify...Ch. 2.2 - The figure shows graphs of f,f,f,andf. Identify...Ch. 2.2 - The figure shows the graphs of three functions....Ch. 2.2 - The figure shows the graphs of four functions. One...Ch. 2.2 - Use the definition of a derivative to find f(x)...Ch. 2.2 - Prob. 52ECh. 2.2 - Prob. 53ECh. 2.2 - a The graph of a position function of a car is...Ch. 2.2 - Prob. 55ECh. 2.2 - Prob. 56ECh. 2.2 - Show that the function f(x)=|x6| is not...Ch. 2.2 - Where is the greatest integer function f(x)=x not...Ch. 2.2 - Prob. 59ECh. 2.2 - a Sketch the graph of the function g(x)=x+|x|. b...Ch. 2.2 - Prob. 61ECh. 2.2 - The left-hand and right-hand derivatives of f at a...Ch. 2.2 - Nick starts jogging and runs faster and faster for...Ch. 2.2 - Prob. 64ECh. 2.2 - Let l be the tangent line to the parabola y=x2 at...Ch. 2.3 - Differentiate the function. f(x)=240Ch. 2.3 - Prob. 2ECh. 2.3 - Differentiate the function. f(x)=5.2x+2.3Ch. 2.3 - Prob. 4ECh. 2.3 - Prob. 5ECh. 2.3 - Differentiate the function. f(t)=1.4t52.5t2+6.7Ch. 2.3 - Prob. 7ECh. 2.3 - Prob. 8ECh. 2.3 - Prob. 9ECh. 2.3 - Differentiate the function. B(y)=cy6Ch. 2.3 - Differentiate the function. F(r)=5r3Ch. 2.3 - Prob. 12ECh. 2.3 - Prob. 13ECh. 2.3 - Prob. 14ECh. 2.3 - Prob. 15ECh. 2.3 - Prob. 16ECh. 2.3 - Differentiate the function. y=x2+4x+3xCh. 2.3 - Prob. 18ECh. 2.3 - Prob. 19ECh. 2.3 - Prob. 20ECh. 2.3 - Prob. 21ECh. 2.3 - Prob. 22ECh. 2.3 - Prob. 23ECh. 2.3 - Prob. 24ECh. 2.3 - Differentiate. f(x)=(5x22)(x3+3x)Ch. 2.3 - Prob. 26ECh. 2.3 - Differentiate. F(y)=(1y23y4)(y+5y3)Ch. 2.3 - Prob. 28ECh. 2.3 - Prob. 29ECh. 2.3 - Prob. 30ECh. 2.3 - Differentiate. y=x2+1x31Ch. 2.3 - Differentiate. y=1t3+2t21Ch. 2.3 - Prob. 33ECh. 2.3 - Prob. 34ECh. 2.3 - Differentiate. y=sss2Ch. 2.3 - Prob. 36ECh. 2.3 - Differentiate. f(t)=t3t3Ch. 2.3 - Prob. 38ECh. 2.3 - Differentiate. F(x)=2x5+x46xx3Ch. 2.3 - Differentiate. A(v)=v2/3(2v2+1v2)Ch. 2.3 - Differentiate. G(y)=BAy3+BCh. 2.3 - Prob. 42ECh. 2.3 - Differentiate. f(x)=xx+cxCh. 2.3 - Prob. 44ECh. 2.3 - Prob. 45ECh. 2.3 - Prob. 46ECh. 2.3 - Prob. 47ECh. 2.3 - Prob. 48ECh. 2.3 - Prob. 49ECh. 2.3 - a Graph the function g(x)=x2/(x2+1) in the viewing...Ch. 2.3 - Prob. 51ECh. 2.3 - Prob. 52ECh. 2.3 - a The curve y=1/(1+x2) is called a witch of Maria...Ch. 2.3 - Prob. 54ECh. 2.3 - Find equations of the tangent line and normal line...Ch. 2.3 - Prob. 56ECh. 2.3 - Prob. 57ECh. 2.3 - Prob. 58ECh. 2.3 - Prob. 59ECh. 2.3 - Prob. 60ECh. 2.3 - Find the first and second derivatives of the...Ch. 2.3 - Prob. 62ECh. 2.3 - The equation of motion of a particle is s=t33t,...Ch. 2.3 - Prob. 64ECh. 2.3 - Prob. 65ECh. 2.3 - The number of tree species S in a given area A in...Ch. 2.3 - Boyles Law states that when a sample of gas is...Ch. 2.3 - Prob. 68ECh. 2.3 - Prob. 69ECh. 2.3 - Prob. 70ECh. 2.3 - Prob. 71ECh. 2.3 - Prob. 72ECh. 2.3 - If f and g are the functions whose graphs are...Ch. 2.3 - Let p(x)=F(x)G(x)andQ(x)=F(x)/G(x), where F and G...Ch. 2.3 - Prob. 75ECh. 2.3 - Prob. 76ECh. 2.3 - Prob. 77ECh. 2.3 - For what values of x does the graph of...Ch. 2.3 - Prob. 79ECh. 2.3 - Prob. 80ECh. 2.3 - Find equations of both lines that are tangent to...Ch. 2.3 - Prob. 82ECh. 2.3 - Prob. 83ECh. 2.3 - Where does the normal line to the parabola y=x21...Ch. 2.3 - Draw a diagram to show that there are two tangent...Ch. 2.3 - a Find equations of both lines through the point...Ch. 2.3 - a Use the Product Rule twice to prove that if f,...Ch. 2.3 - Find the nth derivative of each function by...Ch. 2.3 - Prob. 89ECh. 2.3 - Prob. 90ECh. 2.3 - Prob. 91ECh. 2.3 - Find a parabola with equation y=ax2+bx+c that has...Ch. 2.3 - Prob. 93ECh. 2.3 - A manufacturer produces bolts of a fabric with a...Ch. 2.3 - Prob. 95ECh. 2.3 - The biomass Bt of a fish population is the total...Ch. 2.3 - Let f(x)={x2+1ifx1x+1ifx1 Is f differentiable at...Ch. 2.3 - Prob. 98ECh. 2.3 - Prob. 99ECh. 2.3 - Prob. 100ECh. 2.3 - Prob. 101ECh. 2.3 - Prob. 102ECh. 2.3 - Prob. 103ECh. 2.3 - Let f(x)={x2ifx2mx+bifx2 Find the values of m and...Ch. 2.3 - Prob. 105ECh. 2.3 - Prob. 106ECh. 2.3 - Prob. 107ECh. 2.3 - Draw a diagram showing two perpendicular lines...Ch. 2.3 - Prob. 109ECh. 2.3 - Sketch the parabolas y=x2 and y=x22x+2 Do you...Ch. 2.4 - Differentiate. f(x)=x2sinxCh. 2.4 - Differentiate. f(x)=xcosx+2tanxCh. 2.4 - Differentiate. f(x)=3cotx2cosxCh. 2.4 - Prob. 4ECh. 2.4 - Differentiate. y=sectanCh. 2.4 - Differentiate. g(t)=4sect+tantCh. 2.4 - Differentiate. y=ccost+t2sintCh. 2.4 - Differentiate. y=u(acosu+bcotu)Ch. 2.4 - Prob. 9ECh. 2.4 - Prob. 10ECh. 2.4 - Differentiate. f()=sin1+cosCh. 2.4 - Prob. 12ECh. 2.4 - Differentiate. y=tsint1+tCh. 2.4 - Prob. 14ECh. 2.4 - Differentiate. f()=cossinCh. 2.4 - Prob. 16ECh. 2.4 - Prove that ddx(cscx)=cscxcotx.Ch. 2.4 - Prob. 18ECh. 2.4 - Prove that ddx(cotx)=csc2x.Ch. 2.4 - Prob. 20ECh. 2.4 - Find an equation of the tangent line to the curve...Ch. 2.4 - Prob. 22ECh. 2.4 - Find an equation of the tangent line to the curve...Ch. 2.4 - Prob. 24ECh. 2.4 - a Find an equation of the tangent line to the...Ch. 2.4 - Prob. 26ECh. 2.4 - Prob. 27ECh. 2.4 - Prob. 28ECh. 2.4 - Prob. 29ECh. 2.4 - Prob. 30ECh. 2.4 - Prob. 31ECh. 2.4 - Prob. 32ECh. 2.4 - Prob. 33ECh. 2.4 - Prob. 34ECh. 2.4 - A mass on a spring vibrates horizontally on a...Ch. 2.4 - An elastic band is hung on a hook and a mass is...Ch. 2.4 - A ladder 10 ft long rests against a vertical wall....Ch. 2.4 - An object with weight W is dragged along a...Ch. 2.4 - Find the limit. limx0sin5x3xCh. 2.4 - Prob. 40ECh. 2.4 - Find the limit. limt0tan6tsin2tCh. 2.4 - Prob. 42ECh. 2.4 - Find the limit. limx0sin3x5x34xCh. 2.4 - Prob. 44ECh. 2.4 - Prob. 45ECh. 2.4 - Prob. 46ECh. 2.4 - Prob. 47ECh. 2.4 - Prob. 48ECh. 2.4 - Prob. 49ECh. 2.4 - Prob. 50ECh. 2.4 - Prob. 51ECh. 2.4 - Prob. 52ECh. 2.4 - Find constants A and B such that the function...Ch. 2.4 - Prob. 54ECh. 2.4 - Differentiate each trigonometric identity to...Ch. 2.4 - A semicircle with diameter PQ sits on an isosceles...Ch. 2.4 - The figure shows a circular arc of length .s and a...Ch. 2.4 - Prob. 58ECh. 2.5 - Write the composite function in the form f(g(x))....Ch. 2.5 - Prob. 2ECh. 2.5 - Prob. 3ECh. 2.5 - Prob. 4ECh. 2.5 - Prob. 5ECh. 2.5 - Prob. 6ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 8ECh. 2.5 - Prob. 9ECh. 2.5 - Prob. 10ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 12ECh. 2.5 - Find the derivative of the function. f()=cos(2)Ch. 2.5 - Prob. 14ECh. 2.5 - Find the derivative of the function. h(v)=v1+v23Ch. 2.5 - Find the derivative of the function. f(t)=tsintCh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 18ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 20ECh. 2.5 - Prob. 21ECh. 2.5 - Prob. 22ECh. 2.5 - Prob. 23ECh. 2.5 - Prob. 24ECh. 2.5 - Find the derivative of the function. h()=tan(2sin)Ch. 2.5 - Prob. 26ECh. 2.5 - Find the derivative of the function. y=cosx1+sinxCh. 2.5 - Prob. 28ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 30ECh. 2.5 - Find the derivative of the function. y=cos(sec4x)Ch. 2.5 - Prob. 32ECh. 2.5 - Find the derivative of the function. y=sin1+x2Ch. 2.5 - Prob. 34ECh. 2.5 - Prob. 35ECh. 2.5 - Prob. 36ECh. 2.5 - Find the derivative of the function. y=cot2(sin)Ch. 2.5 - Prob. 38ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 40ECh. 2.5 - Find the derivative of the function. y=x+xCh. 2.5 - Prob. 42ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 44ECh. 2.5 - Find the derivative of the function....Ch. 2.5 - Prob. 46ECh. 2.5 - Find y and y" y=cos(sin3)Ch. 2.5 - Prob. 48ECh. 2.5 - Find y and y" y=1sectCh. 2.5 - Prob. 50ECh. 2.5 - Prob. 51ECh. 2.5 - Prob. 52ECh. 2.5 - Prob. 53ECh. 2.5 - Prob. 54ECh. 2.5 - a Find an equation of the tangent line to the...Ch. 2.5 - a The curve y=|x|/2x2 is called a bullet-nose...Ch. 2.5 - Prob. 57ECh. 2.5 - The function f(x)=sin(x+sin2x),0x, arises in...Ch. 2.5 - Prob. 59ECh. 2.5 - At what point on the curve y=1+2x is the tangent...Ch. 2.5 - Prob. 61ECh. 2.5 - Prob. 62ECh. 2.5 - Prob. 63ECh. 2.5 - Prob. 64ECh. 2.5 - If f and g are the functions whose graphs are...Ch. 2.5 - If f is the function whose graph is shown, let...Ch. 2.5 - If g(x)=f(x), where the graph of f is shown,...Ch. 2.5 - Prob. 68ECh. 2.5 - Prob. 69ECh. 2.5 - Prob. 70ECh. 2.5 - Prob. 71ECh. 2.5 - Prob. 72ECh. 2.5 - Find the given derivative by finding the first few...Ch. 2.5 - Prob. 74ECh. 2.5 - The displacement of a particle on a vibrating...Ch. 2.5 - Prob. 76ECh. 2.5 - Prob. 77ECh. 2.5 - Prob. 78ECh. 2.5 - Prob. 79ECh. 2.5 - Air is being pumped into a spherical weather...Ch. 2.5 - Prob. 81ECh. 2.5 - Prob. 82ECh. 2.5 - Use the Chain Rule to prove the following. a The...Ch. 2.5 - Use the Chain Rule and the Product Rule to give an...Ch. 2.5 - Prob. 85ECh. 2.5 - Prob. 86ECh. 2.5 - Prob. 87ECh. 2.5 - a Write |x|=x2 and use the Chain Rule to show that...Ch. 2.5 - If y=f(u) and u=g(x), where f and g are twice...Ch. 2.5 - Prob. 90ECh. 2.6 - a Find y by implicit differentiation. b Solve the...Ch. 2.6 - a Find y by implicit differentiation. b Solve the...Ch. 2.6 - a Find y by implicit differentiation. b Solve the...Ch. 2.6 - a Find y by implicit differentiation. b Solve the...Ch. 2.6 - Find dy/dx by implicit differentiation. x24xy+y2=4Ch. 2.6 - Find dy/dx by implicit differentiation. 2x2+xyy2=2Ch. 2.6 - Prob. 7ECh. 2.6 - Find dy/dx by implicit differentiation. x3xy2+y3=1Ch. 2.6 - Find dy/dx by implicit differentiation. x2x+y=y2+1Ch. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Prob. 12ECh. 2.6 - Find dy/dx by implicit differentiation. x+y=x4+y4Ch. 2.6 - Prob. 14ECh. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Find dy/dx by implicit differentiation. xy=x2+y2Ch. 2.6 - Find dy/dx by implicit differentiation. xy=1+x2yCh. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Find dy/dx by implicit differentiation....Ch. 2.6 - Prob. 20ECh. 2.6 - If f(x)+x2[f(x)]3=10 and f(1)=2, find f(1).Ch. 2.6 - If g(x)+xsing(x)=x2, find g(0).Ch. 2.6 - Regard y as the independent variable and x as the...Ch. 2.6 - Regard y as the independent variable and x as the...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Prob. 30ECh. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - a The curve with equation y2=5x4x2 is called a...Ch. 2.6 - a The curve with equation y2=x2+3x2 is called the...Ch. 2.6 - Find y by implicit differentiation. x2+4y2=4Ch. 2.6 - Find y by implicit differentiation. x2+xy+y2=3Ch. 2.6 - Find y by implicit differentiation. siny+cosx=1Ch. 2.6 - Find y by implicit differentiation. x3y3=7Ch. 2.6 - If xy+y3=1, find the value of y at the point where...Ch. 2.6 - Prob. 40ECh. 2.6 - Fanciful shapes can be created by using the...Ch. 2.6 - a The curve with equation 2y3+y2y5=x42x3+x2 has...Ch. 2.6 - Prob. 43ECh. 2.6 - Show by implicit differentiation that the tangent...Ch. 2.6 - Find an equation of the tangent line to the...Ch. 2.6 - Prob. 46ECh. 2.6 - Show, using implicit differentiation, that any...Ch. 2.6 - Prob. 48ECh. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Show that the ellipse x2/a2+y2/b2=1 and the...Ch. 2.6 - Prob. 54ECh. 2.6 - a The van der Waals equation for n moles of a gas...Ch. 2.6 - Prob. 56ECh. 2.6 - The equation x2xy+y2=3 represents a rotated...Ch. 2.6 - Prob. 58ECh. 2.6 - Find all points on the curve x2y2+xy=2 where the...Ch. 2.6 - Find equations of both the tangent lines to the...Ch. 2.6 - The Bessel function of order 0, y=J(x) satisfies...Ch. 2.6 - The figure shows a lamp located three units to the...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - A particle moves according to a law of motion...Ch. 2.7 - Graphs of the velocity functions of two particles...Ch. 2.7 - Graphs of the position functions of two particles...Ch. 2.7 - The height in meters of a projectile shot...Ch. 2.7 - If a ball is thrown vertically upward with a...Ch. 2.7 - If a rock is thrown vertically upward from the...Ch. 2.7 - Prob. 10ECh. 2.7 - a A company makes computer chips from square...Ch. 2.7 - a Sodium chlorate crystals are easy to grow in the...Ch. 2.7 - a Find the average rate of change of the area of a...Ch. 2.7 - A stone is dropped into a lake, creating a...Ch. 2.7 - A spherical balloon is being inflated. Find the...Ch. 2.7 - Prob. 16ECh. 2.7 - The mass of the part of a metal rod that lies...Ch. 2.7 - If a tank holds 5000 gallons of water, which...Ch. 2.7 - The quantity of charge Q in coulombs C that has...Ch. 2.7 - Prob. 20ECh. 2.7 - Prob. 21ECh. 2.7 - Some of the highest tides in the world occur in...Ch. 2.7 - Boyles Law states that when a sample of gas is...Ch. 2.7 - Prob. 24ECh. 2.7 - The table gives the population of the world Pt, in...Ch. 2.7 - Prob. 26ECh. 2.7 - Prob. 27ECh. 2.7 - Prob. 28ECh. 2.7 - Prob. 29ECh. 2.7 - The cost function for a certain commodity is...Ch. 2.7 - If px is the total value of the production when...Ch. 2.7 - Prob. 32ECh. 2.7 - The gas law for an ideal gas at absolute...Ch. 2.7 - Invasive species often display a wave of advance...Ch. 2.7 - In the study of ecosystems, predator-prey models...Ch. 2.7 - In a fish farm, a population of fish is introduced...Ch. 2.8 - Prob. 1ECh. 2.8 - a If A is the area of a circle with radius r and...Ch. 2.8 - Each side of a square is increasing at a rate of 6...Ch. 2.8 - Prob. 4ECh. 2.8 - A cylindrical tank with radius 5 m is being filled...Ch. 2.8 - Prob. 6ECh. 2.8 - The radius of a spherical ball is increasing at a...Ch. 2.8 - The area of a triangle with sides of lengths a and...Ch. 2.8 - Suppose y=2x+1, where x and y are functions of t....Ch. 2.8 - Prob. 10ECh. 2.8 - Prob. 11ECh. 2.8 - A particle is moving along a hyperbola xy=8. As it...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - 13-16 a What quantities are given in the problem?...Ch. 2.8 - Prob. 16ECh. 2.8 - Two cars start moving from the same point. One...Ch. 2.8 - Prob. 18ECh. 2.8 - A man starts walking north at 4 ft/s from a point...Ch. 2.8 - A baseball diamond is a square with side 90 ft. A...Ch. 2.8 - The altitude of a triangle is increasing at a rate...Ch. 2.8 - A boat is pulled into a dock by a rope attached to...Ch. 2.8 - At noon, ship A is 100 km west of ship B. Ship A...Ch. 2.8 - Prob. 24ECh. 2.8 - Water is leaking out of an inverted conical tank...Ch. 2.8 - A trough is 10 ft long and its ends have the shape...Ch. 2.8 - A water trough is 10 m long and a cross-section...Ch. 2.8 - A swimming pool is 20 ft wide, 40 ft long, 3 ft...Ch. 2.8 - Gravel is being dumped from a conveyor belt at a...Ch. 2.8 - A kite 100 ft above the ground moves horizontally...Ch. 2.8 - Prob. 31ECh. 2.8 - Prob. 32ECh. 2.8 - The top of a ladder slides down a vertical wall at...Ch. 2.8 - According to the model we used to solve Example 2,...Ch. 2.8 - If the minute hand of a clock has length r in...Ch. 2.8 - Prob. 36ECh. 2.8 - Prob. 37ECh. 2.8 - When air expands adiabatically without gaining or...Ch. 2.8 - Prob. 39ECh. 2.8 - Brain weight B as a function of body weight W in...Ch. 2.8 - Prob. 41ECh. 2.8 - Two carts, A and B, are connected by a rope 39 ft...Ch. 2.8 - Prob. 43ECh. 2.8 - Prob. 44ECh. 2.8 - A plane flies horizontally at an altitude of 5 km...Ch. 2.8 - A Ferris wheel with a radius of 10 m is rotating...Ch. 2.8 - A plane flying with a constant speed of 300 km/h...Ch. 2.8 - Two people start from the same point. One walks...Ch. 2.8 - A runner sprints around a circular track of radius...Ch. 2.8 - The minute hand on a watch is 8 mm long and the...Ch. 2.9 - 14 Find the linearization L(x) of the function at...Ch. 2.9 - Prob. 2ECh. 2.9 - 14 Find the linearization L(x) of the function at...Ch. 2.9 - Prob. 4ECh. 2.9 - Prob. 5ECh. 2.9 - Find the linear approximation of the function...Ch. 2.9 - Prob. 7ECh. 2.9 - Prob. 8ECh. 2.9 - 7-10 Verify the given linear approximation at a=0....Ch. 2.9 - Prob. 10ECh. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - 11-14 Find the differential dy of each function. a...Ch. 2.9 - Prob. 14ECh. 2.9 - 15-18 a Find the differential dy and b evaluate dy...Ch. 2.9 - Prob. 16ECh. 2.9 - Prob. 17ECh. 2.9 - Prob. 18ECh. 2.9 - 19-22 Compute y and dy for the given values of x...Ch. 2.9 - Prob. 20ECh. 2.9 - 19-22 Compute y and dy for the given values of x...Ch. 2.9 - Prob. 22ECh. 2.9 - 23-28 Use a linear approximation or differentials...Ch. 2.9 - Prob. 24ECh. 2.9 - Prob. 25ECh. 2.9 - 23-28 Use a linear approximation or differentials...Ch. 2.9 - Prob. 27ECh. 2.9 - Prob. 28ECh. 2.9 - 29-30 Explain, in terms of linear approximations...Ch. 2.9 - Prob. 30ECh. 2.9 - The edge of a cube was found to be 30 cm with a...Ch. 2.9 - The radius of a circular disk is given as 24 cm...Ch. 2.9 - The circumference of a sphere was measured to be...Ch. 2.9 - Use differentials to estimate the amount of paint...Ch. 2.9 - a Use differentials to find a formula for the...Ch. 2.9 - One side of a right triangle is known to be 20 cm...Ch. 2.9 - Prob. 37ECh. 2.9 - When blood flows along a blood vessel, the flux F...Ch. 2.9 - Establish the following rules for working with...Ch. 2.9 - Prob. 40ECh. 2.9 - Suppose that the only information we have about a...Ch. 2.9 - Suppose that we dont have a formula for g(x) but...Ch. 2.R - Write an expression for the slope of the tangent...Ch. 2.R - Prob. 2CCCh. 2.R - Prob. 3CCCh. 2.R - Define the derivative fa. Discuss two ways of...Ch. 2.R - Prob. 5CCCh. 2.R - Prob. 6CCCh. 2.R - Prob. 7CCCh. 2.R - Prob. 8CCCh. 2.R - Prob. 9CCCh. 2.R - Prob. 10CCCh. 2.R - Give several examples of how the derivative can be...Ch. 2.R - Prob. 12CCCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 2TFQCh. 2.R - Prob. 3TFQCh. 2.R - Prob. 4TFQCh. 2.R - Prob. 5TFQCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 8TFQCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 10TFQCh. 2.R - Prob. 11TFQCh. 2.R - Prob. 12TFQCh. 2.R - Determine whether the statement is true or false....Ch. 2.R - Prob. 14TFQCh. 2.R - Prob. 15TFQCh. 2.R - The displacement in meters of an object moving in...Ch. 2.R - The graph of f is shown. State, with reasons, the...Ch. 2.R - Prob. 3ECh. 2.R - Prob. 4ECh. 2.R - The figure shows the graph of f,f, and f".Identify...Ch. 2.R - Prob. 6ECh. 2.R - The total cost of repaying a student loan at an...Ch. 2.R - Prob. 8ECh. 2.R - Prob. 9ECh. 2.R - Prob. 10ECh. 2.R - Prob. 11ECh. 2.R - Prob. 12ECh. 2.R - Prob. 13ECh. 2.R - Prob. 14ECh. 2.R - 1340 Calculate y. y=x2x+2xCh. 2.R - Prob. 16ECh. 2.R - Prob. 17ECh. 2.R - Prob. 18ECh. 2.R - Prob. 19ECh. 2.R - Prob. 20ECh. 2.R - Prob. 21ECh. 2.R - Prob. 22ECh. 2.R - Prob. 23ECh. 2.R - Prob. 24ECh. 2.R - Prob. 25ECh. 2.R - Prob. 26ECh. 2.R - Prob. 27ECh. 2.R - Prob. 28ECh. 2.R - 1340 Calculate y. sin(xy)=x2yCh. 2.R - Prob. 30ECh. 2.R - Prob. 31ECh. 2.R - Prob. 32ECh. 2.R - Prob. 33ECh. 2.R - Prob. 34ECh. 2.R - Prob. 35ECh. 2.R - Prob. 36ECh. 2.R - Prob. 37ECh. 2.R - Prob. 38ECh. 2.R - Prob. 39ECh. 2.R - Prob. 40ECh. 2.R - If f(t)=4t+1, find f"(2).Ch. 2.R - Prob. 42ECh. 2.R - Find y" if x6+y6=1.Ch. 2.R - Prob. 44ECh. 2.R - Prob. 45ECh. 2.R - Prob. 46ECh. 2.R - Prob. 47ECh. 2.R - Prob. 48ECh. 2.R - Prob. 49ECh. 2.R - Prob. 50ECh. 2.R - Prob. 51ECh. 2.R - Prob. 52ECh. 2.R - Prob. 53ECh. 2.R - Prob. 54ECh. 2.R - Prob. 55ECh. 2.R - How many tangent fines to the curve y=x/(x+1) pass...Ch. 2.R - If f(x)=(xa)(xb)(xc), show that...Ch. 2.R - Prob. 58ECh. 2.R - Prob. 59ECh. 2.R - If f and g are the functions whose graphs are...Ch. 2.R - Prob. 61ECh. 2.R - Prob. 62ECh. 2.R - Prob. 63ECh. 2.R - Prob. 64ECh. 2.R - Prob. 65ECh. 2.R - Prob. 66ECh. 2.R - Prob. 67ECh. 2.R - Prob. 68ECh. 2.R - Prob. 69ECh. 2.R - 6971 Find h in terms of f.and g. h(x)=f(x)g(x)Ch. 2.R - Prob. 71ECh. 2.R - Prob. 72ECh. 2.R - A particle moves on a vertical line so that its...Ch. 2.R - The volume of a right circular cone is V=13r2h,...Ch. 2.R - The mass of part of a wire is x(1+x) kilograms,...Ch. 2.R - The cost, in dollars, of producing x units of a...Ch. 2.R - Prob. 77ECh. 2.R - Prob. 78ECh. 2.R - Prob. 79ECh. 2.R - A waterskier skis over the ramp shown in the...Ch. 2.R - Prob. 81ECh. 2.R - Prob. 82ECh. 2.R - a Find the linearization of f(x)=1+3x3 at a=0....Ch. 2.R - Prob. 84ECh. 2.R - A window has the shape of a square surmounted by a...Ch. 2.R - 8688 Express the limit as a derivative and...Ch. 2.R - Prob. 87ECh. 2.R - Prob. 88ECh. 2.R - Prob. 89ECh. 2.R - Prob. 90ECh. 2.R - Prob. 91ECh. 2.R - Show that the length of the portion of any tangent...Ch. 2.P - Find points P and Q on the parabola y=1x2 so that...Ch. 2.P - Prob. 2PCh. 2.P - Show that the tangent lines to the parabola...Ch. 2.P - Prob. 4PCh. 2.P - Prob. 5PCh. 2.P - Prob. 6PCh. 2.P - Prob. 7PCh. 2.P - Prob. 8PCh. 2.P - Prob. 9PCh. 2.P - Prob. 10PCh. 2.P - How many lines are tangent to both of the circles...Ch. 2.P - If f(x)=x46+x45+21+x, calculate f(46)(3). Express...Ch. 2.P - The figure shows a rotating wheel with radius 40...Ch. 2.P - Tangent lines T1 and T2 are drawn at two points P1...Ch. 2.P - Let T and N be the tangent and normal lines to the...Ch. 2.P - Prob. 16PCh. 2.P - Prob. 17PCh. 2.P - Let P(x1,y1) be a point on the parabola y2=4px...Ch. 2.P - Suppose that we replace the parabolic mirror of...Ch. 2.P - Prob. 20PCh. 2.P - Prob. 21PCh. 2.P - Given an ellipse x2/a2+y2/b2=1, where ab, find the...Ch. 2.P - Prob. 23PCh. 2.P - Prob. 24PCh. 2.P - A lattice point in the plane is a point with...Ch. 2.P - Prob. 26PCh. 2.P - Prob. 27PCh. 2.P - Prob. 28P
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY