The function f has an antiderivative, F, and F \left( 3 \right) = 4.
Express \int_{3}^{x} f \left( t \right) dt in terms of F and x.
Hence, calculate \dfrac{d}{d x} \int_{3}^{x} f \left( t \right) dt.
Consider the function f \left( t \right) = - 4 t.
Calculate \int_{6}^{x} f \left( t \right) dt.
Hence, calculate \dfrac{d}{d x} \int_{6}^{x} f \left( t \right) dt.
Consider \dfrac{d}{d x} \int_{6}^{x} f \left( t \right) dt, and explain the result of finding the derivative in terms of x, of an integral function expressed in terms of t.
Consider the function f \left( t \right) = 12 t + 9.
Calculate \int_{ - 3 }^{x} f \left( t \right) dt.
Hence, calculate \dfrac{d}{d x} \int_{ - 3 }^{x} f \left( t \right) dt.
Calculate the following:
\dfrac{d}{d x} \int_{2}^{x} \left( 6 t^{2} - 8 t + 3\right) dt
\dfrac{d}{d x} \int_{ - 2 }^{x} \left(\sqrt{t + 3} - 6 t\right) dt
\dfrac{d}{d x} \int_{0}^{x} \left(t^{\frac{2}{3}} - t^{2}\right) dt
\dfrac{d}{d x} \int_{10}^{x} \left( - 4 t^{3} + 4 t - 7\right) dt
\dfrac{d}{d x} \int_{ - 6 }^{x} \dfrac{2^{t} t^{5}}{3} dt
\dfrac{d}{d x} \int_{13}^{x} \left(\dfrac{3}{t^{2}} - \dfrac{4}{t^{3}}\right) dt
Consider the expression \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}} dt.
State the restrictions on the value of k, for the integration to be possible.
Evaluate \int_{k}^{x} \dfrac{1}{\sqrt{t}} dt.
Hence, evaluate \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}} dt.
Evaluate the following:
\int_{1}^{4} \dfrac{d}{d x}\left(\sqrt{x} + 2 x^{2}\right) dx
\int_{0}^{\pi} \dfrac{d}{d x} \cos \left( 2 x\right) dx
\int_{ - 1 }^{0} \dfrac{d}{d t}\left(e^{ - 2 t } + t\right) dt
\int_{2}^{5} \dfrac{d}{d x}\left( 3 x^{2} + 2 x - 1\right) dx
\int_{0}^{\frac{\pi}{2}} \dfrac{d}{d x} \sin \left( 3 x - \dfrac{\pi}{4}\right) dx
\int_{2}^{5} f' \left( x \right) dx given f \left( 2 \right) = 4, f \left( 5 \right) = 12
Given that F \left( x \right) = \int_{1}^{x} f \left( t \right) dt, with F \left( 4 \right) = 78 and F'' \left( x \right) = 6 x, find f \left( t \right).
The increments formula to estimate the change in y is:
\delta y \approx {f}'\left(x \right )\times\delta xUse the increments formula to estimate the change in F \left( x \right) as x changes from 5 to 5.01. The function F is defined by F \left( x \right) = \int_{0}^{x} \cos \left(t^{2}\right) dt. Round your answer to three decimal places.
A function F is defined by F \left( x \right) = \int_{0}^{x} \dfrac{5}{\sqrt{t^{2} + 7}} dt. Use the increments formula to estimate the change in F \left( x \right) as x changes from 3 to 3.5. Round your answer to three decimal places.
A function F is defined by F \left( x \right) = \int_{0}^{x} \left( 2 t^{3} - 11\right)^{2} dt. Use the increments formula to estimate the change in F \left( x \right) as x changes from 2 to 2.01. Round your answer to two decimal places.
Consider the function f \left( t \right) = 2 t, where t \geq 0. Let A \left( x \right) be the area function which represents the area bound by f \left( t \right) and the horizontal axis from 0 to x. That is,
A \left( x \right) = \int_{0}^{x} f \left( t \right) dtFind an expression for A \left( x \right).
Hence determine the area under the function f \left( t \right), from t = 0 to t = 4.
Consider the function f \left( t \right) = 2 t - 6, and the signed area function A \left( x \right) = \int_{0}^{x} f \left( t \right) dt.
Complete the following table of values:
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| A(x) | 0 | -5 | -5 | 7 |
State the range of values of x for which the function A \left( x \right) is increasing.
Sketch a graph of y = A \left( x \right), for 0 \leq x \leq 7.
Determine the values of x where A \left( x \right) = 0.
Explain the significance of these values of x.
Write the function A \left( x \right), in terms of x.