X 951961..: "Towards a winter tra X + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/higherOrderDerivatives1 H Men Custom Suits... OBAMASENSE.. O BREEZ($ 20... b Big Dude USA - Bi... 20 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 4 hours Get Help Saved! Erase Edit Qia 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 21 22 Higher order derivatives Exercise. Let f(x) = 24 - 2x3 + 7x - 7. Compute the following derivatives. f' (2) = f" (20 ) = f 'll ( 2 ) = filll ( 2c ) = f (5) ( 20 ) = f (50) ( 2 ) = f (5000) (2) = X fcafa5..: "As well as a magenta X + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/14_Q4-1151_ex_1 H Men Custom Suits... O BRASEXX. 0 9:86218 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 3 hours Get Help Save Erase Edit Qia 13 14 15 16 18 19 20 21 22 Higher order derivatives Exercise. On planet X a stone is thrown vertically into the air at an initial velocity of 60 ft/s. The height s (in feet) of the stone above the ground Reveal Hint after t seconds is s(t) = 64 + 60t - 4t2. (a) Find the velocity function v and the acceleration function a. v(t) = a(t) = (b) With what speed will the stone strike the ground? ANSWER: The stone will strike the ground with the speed ft/s - Previous Next ->X 3e0d77...: "Near a stormy tree' X + ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/velocityFromGraph1 ... H Men Custom Suits... O BABEXX.. @ :RE($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Exercise. An object begins moving along a vertical line at time t = 0s and stops at time t = 6s. Its height above the ground at time t is given by h(t) depicted in the graph below, where h is measured in meters. h 6 4 3 2 t 2 3 4 5 6 The average velocity of the object on the interval [0, 6] is Vav = ? m/s. Choose the best description of the shape of the graph on the time interval (0, 1). Increasing and concave upX 3e0d77..: "Near a stormy tree' X + ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/velocityFromGraph1 0 H Men Custom Suits... O BASEXX.. @ SHEHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 3 hours Get Help Save Erase Edit Qia Choose the best description of the shape of the graph on the time interval (0, 1). Increasing and concave up Increasing and concave down Decreasing and concave up Decreasing and concave down Correct The object is closest to the ground at t = s. The velocity when the object is closest to the ground is V = ? m /s . At which time is the velocity greatest? 0s 2s 3s 4s Correct Estimated as an integer, the maximum velocity is V = ? m/s.X 3e0d77...: "Near a stormy tree' X + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/velocityFromGraph1 0 H Men Custom Suits... OBAMASENSE. O BERHERS 20... b Big Dude USA - Bi... 30 DOKV a Amazon.com: OA... a DXL Twenty-Eight.. a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia 6s Correct Estimated as an integer, the maximum velocity is v = ? m / s . Let v(t) be the velocity of the object at time t. Choose the greatest value amongst the following choices. v(4.5) v (4.6) v(4.7) v (4.8) Correct The time interval on which the velocity is increasing is At which of the following times is the speed greatest? Is 2s 3s As Correct - Previous Next ->X ce2c05..: "Opposite of the use x + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/velocityFromGraph2 0 ... H Men Custom Suits... O SHAREXX... O SHERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA O in 3 hours Get Help Save Erase Edit Qia Exercise. An object begins moving along a horizontal line at time t = 0s and stops at time t = 6s. Its position from the origin at time t is given by s(t) depicted in the graph below, where s is measured in meters. 5 4 3 2 1 t 1 2 3 4 6 The average velocity of the object on the interval [0, 6] is Vav = ? m/s.X 3da4d6..: "Above the super lin X + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/rateOfChange1 H Men Custom Suits... O BRASEX... O SHERHAS 20... [ Big Dude USA - Bi... 30 DOKV a Amazon.com: OA... a a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Exercise. An oil tank is to be drained for cleaning. There are V gallons of oil left in the tank t minutes after the draining has begun, where V(t) = 45(60 - t)2. The average rate at which the oil drains in the time interval [0, 15] is AR = ? gal/min. The average rate at which the oil drains in the time interval [10, 15] is AR = ? gal/min. The rate at which the oil drains 15 minutes after draining has begun is R = ? gal/min. The average rate at which the oil drains during the time interval [15, 15 + h] for 0 X 0205c5..: "With the tall angle" X + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/exponentialReview3 H Men Custom Suits... BASE.. @ :HR($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 3 hours Get Help Save Erase Edit Qia 12 13 14 15 16 19 20 21 22 Higher order derivatives Exercise. Let f(x) = 10g3 (x - 9). The domain of f is The average rate of change of f between the x = 10 and x = 18 is The domain of f-1 is The inverse function to f is f-1 (20 ) = - Previous Next >X 8e6do0...: "Toward the good re X + -> C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/functionReview2 0 H Men Custom Suits... O SHAREXX... . SHEHAS 20... b Big Dude USA - Bi... 20 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Exercise. The (entire) graph of a function f is given below. y 6 The domain of f is O The range of f (from bottom to top) is J JX 8e6do0...: "Toward the good re X + -> C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/functionReview2 0 H Men Custom Suits... O BABE. O BERHERS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia f is one-to-one. True False Correct The domain of f - (from left to right) is U The range of f-1 is f ( 5) = f ( 2) = f(0) = f-1 (7 ) = f -' ( 6) = f- (-3) = ?X e2bc65..: "As opposed to the | X + - C ximera.osu.edu/mooculus/higherOrderDerivativesAndGraphs/exercises/exerciseList/higherOrderDerivativesAndGraphs/exercises/functionReview3 H Men Custom Suits... OBHARENA.. O SHERHERS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 3 hours Get Help Save Erase Edit Qia 12 13 14 15 16 19 20 21 22 Higher order derivatives Exercise. Consider the function f(x) = va + 2. The inverse to f is f-1 (20 ) = - Previous Next -> Courses About Social Built at The Ohio State University Calculus One FAQ Facebook f with support from NSF Grant DUE- Calculus Two Development Team Twitter y 1245433, the Shuttleworth Calculus Three Workshop Google Plus G+ Foundation, the Department of Contact Us GitHub Mathematics, and the Affordable Learning Exchange. THE OHIO STATE UNIVERSITY 2013-2023, The Ohio State University - Ximera team 100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210-1174 Phone: (773) 809-5659 | ContactX Standard form - Ximera X + -> C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/breakGround H Men Custom Suits... BABEXX.. . SHEHAS 20... Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Implicit $1 Standard $1.1 Implicit $1.2 $1.3 differentiation form differentiation Derivatives of inverse Two young In this section we exponential mathematicians differentiate functions Abstract. Two young mathematicians discuss the standard form of a line. Check out this dialogue between two calculus students (based on a true story): Devyn Riley, I think we've been too explicit with each other. We should try to be more implicit. Riley I. Um. Don't really. .. Devyn I mean when plotting things! Riley Okay, but I still have no idea what you are talking about. Devyn Remember when we first learned the equation of a line, and the "standard form" was ax + by = c or something, which is totally useless for graphing. Also a circle is 2 2 + y2 = 72 or something, and here y isn't even a function of x. Riley Ah, I'm starting to remember. We can write the same thing in two ways. For example, if you write y = mx + b, then y is explicity a function of x but if you write ax + by = c,X Standard form - Ximera X + -> C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/breakGround H Men Custom Suits... O BRASEX.. @ SHEERS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 3 hours Get Help Save Erase Edit Qia or something, and here y isn't even a function of x. Riley Ah, I'm starting to remember. We can write the same thing in two ways. For example, if you write y = mx + b, then y is explicity a function of x but if you write ax + by = c, then y is implicitly a function of x. Devyn What I'm trying to say is that we need to learn how to work with these "implicit" functions. Problem. Consider the unit circle Reveal Hint a ty? = 1. The point P = (0, 1) is on this circle. Reason geometrically to determine the slope of the line tangent to x2 + y = 1 at P. The slope is Problem. Consider the unit circle Reveal Hint a2 + y' = 1. The point 2 2 ) is on this circle. Reason geometrically to determine the slope of the line tangent to x2 + y2 = 1 at P. The slope is - Previous Next ->X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation H Men Custom Suits... O SHAREXX.. O SHERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA @ in 3 hours Get Help Save Erase Edit Qia Implicit $1 Standard $1.1 Implicit $1.2 $1.3 differentiation form differentiation Derivatives of inverse Two young In this section we exponential mathematicians differentiate functions Abstract. In this section we differentiate equations that contain more than one variable on one side. Review of the chain rule Implicit differentiation is really just an application of the chain rule. So recall: Theorem. (Chain Rule) If f(x) and g(x) are differentiable, then a f (g (z) ) = f'(9(z))9' ( 2 ) . Of particular use in this section is the following. If y is a differentiable function of x and if f is a differentiable function, then da (f(y)) = f'(y) - da ( y ) = f' ( 2 ) do . Implicit differentiation The functions we've been dealing with so far have been defined explicitly in terms of the independent variable. For example: x- 2 y = 3x - 2x + 1, y = esx, y = 22 - 3x + 2 However, this is not always necessary or even possible to do. Sometimes we choose to or we have to define a function implicitly . In this case, the dependent variable is not stated policitly in terms of an independent variable. Some examples arX Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation ... H Men Custom Suits... BASEXX... @ :3HR($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia However, this is not always necessary or even possible to do. Sometimes we choose to or we have to define a function implicitly . In this case, the dependent variable is not stated explicitly in terms of an independent variable. Some examples are: aty = 4, *3 + y' = gxy , 24 + 3202 = 2 2/3 + y 2/3 + 1 . Your inclination might be simply to solve each of these equations for y and go merrily on your way. However, this can be difficult or even impossible to do. Since we are often faced with a problem of computing derivatives of such functions, we need a method that will enable us to compute derivatives of implicitly defined functions. We'll start with a basic example. Example. Consider the curve (a circle) defined by: (a) Find the slope of the line tangent to the circle at the point (2, 2). Explanation. y y = V1 - x2X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BASEXX... . SHEHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Explanation. The curve defined by the equation x2 + y2 = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = 1 - x2 andy = -V1 -x2. The point (12, lies on the graph of the function f(x) = V1 - 2. Let's compute the derivative of f. f' (z) = 2V1 -202 Therefore, the slope of the tangent line at the point (2, 2 ) is given byX Implicit differentiation - Ximera X + - C oximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BRASEX... @ SHEHAS 20... b Big Dude USA - Bi... 70 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia f' ( 2 ) = 2V1 -22 Therefore, the slope of the tangent line at the point (2, 2 ) is given by slope = f' (_2 ) (b) Find the slope of the line tangent to the circle at the point (12, -12 ) Explanation. y y = -X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... OSHASEX... O SHERHAS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XINERA in 3 hours Get Help Save Erase Edit Qia (b) Find the slope of the line tangent to the circle at the point (2, - Explanation. y The curve defined by the equation x2 + y2 = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = 1 - x2 and y = -V1 - x2. The point 2, -2 lies on the graph of the function f(x) = -V1 -12. Let's compute the derivative of f. ? f'(2) 2V1 -22X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BABE... @ SHAHR($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Therefore, the slope of the tangent line at the point (2, -12 ) is given by slope = f' ( 2 2 ) - 2 Notice that we had to differentiate twice, not to mention that we had to first solve for y in terms of x in order to compute these two slopes. Let's take a different approach, namely let's use implicit differentiation. Example. Consider the curve (a circle) defined by: ty = 1 (a) Compute dy . (b) Find the slope of the line tangent to the circle at (2, 2). (c) Find the slope of the line tangent to the circle at (2, -2). Explanation. The curve defined by the equation x2 + y" = 1 is not a graph of a function. If we solve for y, we obtain two solutions: y = v1 - x2 andy = -V1 - x2. Therefore, we can say that any point (x, y) on the curve lies on the graph of some function f. Starting with 202 + y? = 1 we differentiate both sides of the equation with respect to x to obtain da (2 2 + y2 ) = d Applying the sum rule we see d -2 2 + = 0.X Implicit differentiation - Ximera X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInImplicitDifferentiation 0 H Men Custom Suits... O BASEX... @ 9:86218 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia Applying the sum rule we see d - 72 d da da = 0. Let's examine each of these terms in turn. To start d - 202 da ? On the other hand, y2 is somewhat different. Here we assume that y = f(x) for some function f, defined on some open interval (this is true for all points -1 ?X Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions H Men Custom Suits... O BRASEX.. @ 3:HR($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia One may suspect that we can use the fact that de" = et, to deduce the derivative of In(x). We will use implicit differentiation to exploit this relationship computationally. Theorem. (The Derivative of the Natural Logrithm) d In ( 20 ) = dx Explanation. Recall In(x) = y exactly when ey = x and * > Hence es = x _y = Differentiate both sides. ag - 1 Implicit differentiation. dy 1 ? dy py Solve for dx Since y = In(x), 4 In(x) = Question. Compute: dx d ( - In(cos(z)) ) = From the derivative of the natural logarithm, we can deduce another fact:X Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions 0 90 H Men Custom Suits... O BABE.. 0 9:28x($ 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight.. a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA @ in 3 hours Get Help Save Erase Edit Qia From the derivative of the natural logarithm, we can deduce another fact: Theorem. (The derivative of any logarithm) Let b be a positive real number. Then d log% (2 ) = - 1 dx a In (b) Explanation. Here we need to remember that In (a) log (z ) = So we may write d log% ( 20) = - d In(x dx dx 1 d In(x) In(b) dx Question. Compute: d 1087 ( 20 ) = dx We can also compute the derivative of an arbitrary exponential function.X Derivatives of inverse exponen X + - C ximera.osu.edu/mooculus/calculus1TextbookBySection/implicitDifferentiation/implicitDifferentiation/digInDerivativesOfInverseExponentialFunctions 0 90 H Men Custom Suits... SHARES.. O RRHHRS 20... b Big Dude USA - Bi... 50 DOKV a Amazon.com: OA... a DXL Twenty-Eight... a Amazon.com: DXL a Amazon.com: ZHILI a Amazon.com: Car... XIMERA in 3 hours Get Help Save Erase Edit Qia We can also compute the derivative of an arbitrary exponential function. Theorem. (The derivative of an exponential function) dac -a" = a" . In(a). Explanation. Here we need to be slightly sneaky. Note a" = en(a") = ex In(a) So we may write d d r In(a) dx da = ex In(a) . Question. Compute: -72 = - Previous Next >