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4.08 The fundamental theorem of calculus

Worksheet
Integration and differentiation as opposite processes
1

The function f has an antiderivative, F, and F \left( 3 \right) = 4.

a

Express \int_{3}^{x} f \left( t \right) dt in terms of F and x.

b

Hence, calculate \dfrac{d}{d x} \int_{3}^{x} f \left( t \right) dt.

2

Consider the function f \left( t \right) = - 4 t.

a

Calculate \int_{6}^{x} f \left( t \right) dt.

b

Hence, calculate \dfrac{d}{d x} \int_{6}^{x} f \left( t \right) dt.

c

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.

3

Consider the function f \left( t \right) = 12 t + 9.

a

Calculate \int_{ - 3 }^{x} f \left( t \right) dt.

b

Hence, calculate \dfrac{d}{d x} \int_{ - 3 }^{x} f \left( t \right) dt.

4

Calculate the following:

a

\dfrac{d}{d x} \int_{2}^{x} \left( 6 t^{2} - 8 t + 3\right) dt

b

\dfrac{d}{d x} \int_{ - 2 }^{x} \left(\sqrt{t + 3} - 6 t\right) dt

c

\dfrac{d}{d x} \int_{0}^{x} \left(t^{\frac{2}{3}} - t^{2}\right) dt

d

\dfrac{d}{d x} \int_{10}^{x} \left( - 4 t^{3} + 4 t - 7\right) dt

e

\dfrac{d}{d x} \int_{ - 6 }^{x} \dfrac{2^{t} t^{5}}{3} dt

f

\dfrac{d}{d x} \int_{13}^{x} \left(\dfrac{3}{t^{2}} - \dfrac{4}{t^{3}}\right) dt

5

Consider the expression \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}} dt.

a

State the restrictions on the value of k, for the integration to be possible.

b

Evaluate \int_{k}^{x} \dfrac{1}{\sqrt{t}} dt.

c

Hence, evaluate \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}} dt.

6

Evaluate the following:

a

\int_{1}^{4} \dfrac{d}{d x}\left(\sqrt{x} + 2 x^{2}\right) dx

b

\int_{0}^{\pi} \dfrac{d}{d x} \cos \left( 2 x\right) dx

c

\int_{ - 1 }^{0} \dfrac{d}{d t}\left(e^{ - 2 t } + t\right) dt

d

\int_{2}^{5} \dfrac{d}{d x}\left( 3 x^{2} + 2 x - 1\right) dx

e

\int_{0}^{\frac{\pi}{2}} \dfrac{d}{d x} \sin \left( 3 x - \dfrac{\pi}{4}\right) dx

f

\int_{2}^{5} f' \left( x \right) dx given f \left( 2 \right) = 4, f \left( 5 \right) = 12

7

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).

8

The increments formula to estimate the change in y is:

\delta y \approx {f}'\left(x \right )\times\delta x

Use 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.

9

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.

10

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.

The signed area function
11

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) dt
a

Find an expression for A \left( x \right).

b

Hence determine the area under the function f \left( t \right), from t = 0 to t = 4.

12

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.

a

Complete the following table of values:

x01234567
A(x)0-5-57
b

State the range of values of x for which the function A \left( x \right) is increasing.

c

Sketch a graph of y = A \left( x \right), for 0 \leq x \leq 7.

d

Determine the values of x where A \left( x \right) = 0.

e

Explain the significance of these values of x.

f

Write the function A \left( x \right), in terms of x.

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Outcomes

3.2.15

examine the concept of the signed area function F(x)=∫ {from a to x} f(t)dt

3.2.16

apply the theorem F'(x)=d/dx ( ∫ {from a to x} f(t)dt)=f(x), and illustrate its proof geometrically

3.1.10

use the increments formula: δy≅dy/dx × δx to estimate the change in the dependent variable y resulting from changes in the independent variable x

3.2.17

develop the formula ∫ {from a to b} f(x)dx= F(b)−F(a) and use it to calculate definite integrals

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