Posts tagged calc i
How and when to use L'Hospital's rule

L’Hospital’s Rule is used to get you out of sticky situations with indeterminate limit forms. If you plug in the number you’re approaching to the function for which you’re trying to find the limit and your result is one of the indeterminate forms above, you should try applying L’Hospital’s Rule.

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Related rates problems with inflating and deflating balloons

To solve a related rates problem, complete the following steps: 1) Construct an equation containing all the relevant variables. 2) Differentiate the entire equation with respect to (time), before plugging in any of the values you know. 3) Plug in all the values you know, leaving only the one you’re solving for. 4) Solve for your unknown variable.

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Optimization problems with an open-top box

For example, these are all things we can find by applying the optimization process to the real world: the dimensions of a rectangle that maximize or minimize its area or perimeter, the maximum product or minimum sum of squares of two real numbers, the time at which velocity or acceleration is maximized or minimized, the dimensions that maximize or minimize the surface area or volume of a three-dimensional figure, the production or sales level that maximizes profit, etc.

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Solving vertical motion problems

Vertical motion is any type of upwards or downwards motion that is constant. In a vertical motion problem, you may be asked about instantaneous velocity, and/or average velocity. To solve for instantaneous velocity we will need to take the derivative of our position function.

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Using the quotient rule to find the derivative

Just as we always use the product rule when two variable expressions are multiplied, we always use the quotient rule whenever two variable expressions are divided. So to find the derivative of a quotient, we use the quotient rule.

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Derivatives of inverse hyperbolic functions

To build our inverse hyperbolic functions, we need to know how to find the inverse of a function in general. To find the inverse of a function, we reverse the x and the y in the function. So for y=cosh(x), the inverse function would be x=cosh(y).

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Modeling sales decline with exponential equations

In order to model sales decline with the exponential decay equation, the decline must have a constantly and exponentially rate of decline. If it does, we can use our standard exponential change equation.

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Solving limits with factoring

If you tried to solve the limit with substitution and it didn’t work, factoring should be the next thing you try. The goal will be to factor the function, and then cancel any removable discontinuities, in order to simplify the function, so that it can be evaluated.

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Finding derivatives of logs and natural logs

The derivatives of base-10 logs and natural logs follow a simple derivative formula that we can use to differentiate them. With derivatives of logarithmic functions, it’s always important to apply chain rule and multiply by the derivative of the log’s argument.

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Finding the equation of the tangent line at a point

When a problem asks you to find the equation of the tangent line, you’ll always be asked to evaluate at the point where the tangent line intersects the graph. You’ll need to find the derivative, and evaluate at the given point.

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