Derivative of a function : 11th grade math worksheets with answers.

11th grade math worksheets

Math worksheets on the derivative of a function and the graphical interpretation of the derivative number in 11th grade with detailed corrections.

Exercise 1:

Derive the function f in the following cases:

1. $\,f(x)=-4x^3+2x^2-3x+1\,.$

2. $\,f(x)=\frac{3x^2-4x+2}{2}\,.$

3. $\,f(x)=(\sqrt{x+1})\times \,(x^2-2)\,.$

4. $\,f(x)=(2x-\sqrt{x})\times \,(x+4)\,.$

5. $\,f(x)=\frac{1}{1-4x}\,.$

6. $\,f(x)=\frac{-3}{2x-1}\,.$

7. $\,f(x)=\frac{2x-1}{3x+2}\,.$

8. $\,f(x)=\frac{3x^2-4x+1}{2x-3}\,.$

9. $\,f(x)=(5x^2+1)^2\,.$

10. $\,f(x)=(-2x-1)^3\,.$

11. $\,f(x)=\sqrt{3x-4}.$

12. $\,f(x)=2x\sqrt{-3x+2}.$

Exercise 2:

Determine the equation of the tangent T to the representative curve of the function f at the point of abscissa a in the following cases:

1. f(x)= 3x²-x+1 with a= -1.

2. $\,f(x)=\frac{2x-1}{x-2}\,$ with a= 3.

3. $\,f(x)=\frac{\sqrt{x}}{x}\,$ with a= 9.

Exercise 3:

Let f be the function defined on $\mathbb{R^*}\,$ by :

$\,f(x)=\frac{-x^2+2x-1}{x}$ .

We note C its representative curve in an orthonormal frame.
1. Determine the abscissae of the points on the curve C where the tangent is horizontal.
2. Are there points on the curve C where the tangent has a directrix equal to – 2 ?

3 Determine the abscissa of the points on the curve C where the tangent is parallel to the line of equation $\,y=-\,\frac{2}{3}x-5$ .

Exercise 4 – Equation of the tangent to a representative curve

Let f be the function defined on R by $f(x)\,=\,x^4\,-2x\,+\,1$ .
Let (Cf ) be its representative curve.

1. Give, with justification, the equation of the tangent (T) to the curve (Cf ) at point A of abscissa 0.

2. Draw in the same frame of reference the curve (Cf ) and the tangent (T) on the interval [- 1 ; 1,5].

Exercise 5 – Calculating a limit

The purpose of this exercise is to calculate the following limit:

$\lim_{h\,\to\,0}\frac{(1+h)^{2005}-1}{h}$ .

For this we consider the function defined on $\mathbb{R}$ by $f(x)=(1+x)^{2005}$ .

1. Calculate the derivative f’ of the function f. Calculate f ‘ (0).

2. Calculate the average growth of the function f between 0 and h. Deduce the above limit.

Exercise 6 – Cost price and speed of a truck

A truck has to travel 150 km.
Its diesel consumption is $6+\frac{v^2}{300}$ llitres per hour, where is its speed in.
The price of diesel is 0,9 € per liter and the heater is paid 12 € per hour.

1. Let t be the duration of the journey in hours. Express t as a function of the speed .

2. Calculate the cost price P(v) of the trip as a function of v.

3. What must be the speed v of the truck so that the cost price P(v) of the run is minimal?

Exercise 7 – Vertex of a parabola

Let (P) be the parabola defined by the function $f(x)\,=\,x^2\,-\,3x\,+\,1$ .
Compute the coordinates of its vertex S.

Exercise 8 – Study of a rectangle

Consider a rectangle whose perimeter P is equal to 4 cm.

1. Determine its dimensions (length L and width l) knowing that its area S is equal to $\frac{3}{4}$ cm².

2. We now search for the dimensions of the rectangle so that its area S is maximal.

a. Express S as a function of the width l.

b. Consider the function f defined on $\mathbb{R}$ by .

Calculate the derivative f’ of f and study its sign.

Draw the table of variations of the function f.

Draw the graph (Cf ) of the function f on [0 ; 2].

c. Deduce the dimensions of the rectangle whose perimeter P is equal to 4 m and whose area S is maximal.

Exercise 9 – Numerical function and root

Consider the function f defined on R by: $f(x)\,=\,x^3\,-\,3x\,-\,3$ .
We note (Cf ) its graphical representation.

Calculate the derivative f ‘ of f and study its sign.

2. Draw up the table of variations of the function f.

3. Determine an equation of the tangent (T) to (Cf ) at the point of abscissa 0.

4. Draw (T) and (Cf ) in the same reference frame.

5. Prove that the equation f(x) = 0 has a unique solution $\alpha$ in the interval [2 ; 3].

6. Give an approximate value of $\alpha$ , to the nearest $10^{-1}$ .

Exercise 10 – Variation table and equation

1. Draw up the table of variations of the function f defined on R by : $f(x)\,=\,x^2\,-3x\,+\,2.$
2. Solve the equation f(x) = 0.

Exercise 11 – Study of two functions and tangents

Consider the function defined by .
We note (Cf ) its representative curve.
Consider also the function g defined by g(x) = 3 – x.
We note (D) its graphical representation.

1. Calculate the derivative f’ of f.

2. Determine an equation of the tangent (T) to the curve (Cf ) at the point of abscissa .

3. Solve by calculation the equation g(x) = f(x).

4. Specify the coordinates of the intersection points of (Cf ) and (D).

5. Draw the lines (T), (D) and the curve (Cf ) on the same reference frame.

Exercise 12 – Determining the derivative of numerical functions

Derive the following functions:

$f(x)=4x^2-3x+1\\g(x)=(2x+3)(3x-7)\\h(x)=\frac{2x+4}{3x-1}\,pour\,x\neq\frac{1}{3}\\k(x)=(2x^2+3x+1)^2$

Exercise 13 – Derivative of several functions

Derive the following functions:

$f(x)=x^2\\g(x)=3x^4-2x^3+5x-4\\h(x)=\sqrt{x}(1-\frac{1}{x})\\k(x)=\frac{x+5}{x^2+1}$

Exercise 14 – Absolute value and derivability

Let be a function defined on $\mathbb{R}$ by $f(x)=\,%7C\,2x+3\,\,%7C$ .

Study the derivability of on $\mathbb{R}$ .

Exercise 15 – Derivative of a power function

Prove that if u is a differentiable function on an interval I, then:

a) ^u2 is differentiable on I and (^u2)’=2uu’.

b) ^u3 is differentiable on I and (^u3)’^=3u2u‘.

Exercise 16 – Sense of variation

Consider the function f defined by on $\mathbb{R}$ .

1. Prove that $f(x)\leq\,,\frac{1}{4}$ for all x belonging to $\mathbb{R}$ .

2. Deduce that the function f has a maximum at $x=\frac{1}{2}$ .

3. Show that $f(x)=\frac{1}{4}-,(,x-\frac{1}{2},,)^2$ .

4. Deduce that the function f is increasing on the interval $%5D-\infty;\frac{1}{2}%5B$ and decreasing on $%5D\frac{1}{2};+\infty%5B$ .

Exercise 17

Let $\,f$ be the function defined on $\,\mathbb{R}$ by $\,f(x)=x^2+6x+5$

1. Study the variations of $\,f$ on $\,\mathbb{R}$ .

2. Determine the coordinates of the points of intersection between the representative curve of $\,f$ and the line $\,D$ of equation $\,y=\frac{1}{2}x-2$ .

Exercise 18

Study the variations on $\,\mathbb{R}$ of the function f defined by $\,f(x)=3x-4x^3$ .

Exercise 19

Let f be the function defined on $\,\mathbb{R}$ by :

$\,f(x)=\frac{-4x-4}{x^2+2x+5}$ .

1. Study the variations of f on $\,\mathbb{R}$ .

2. Determine the coordinates of the point A, intersection between the curve representing f and the x-axis.

3. Determine an equation of the tangent T to the representative curve of $\,f$ at point A.

Exercise 20

Study the variations on ]-2 ; 1[ of the function $\,f$ defined by :

$\,f(x)=\frac{-5x^2+4x-8}{x^2+x-2}$ .

Exercise 21 – Representative curve, derivative and tangent

Let be the function defined on $\mathbb{R}$ by $f(x)\,=\,\frac{1}{4}x^4\,-2x^2\,+\,3$

Its graphical representation in an orthonormal frame of reference is called .

1) a) Study the parity of . What can we deduce from this for ?

b) Determine the expression of the derivative function of and derive the table of variation of

2) a) Determine an equation of the tangent at at the point of abscissa 1.

b) This tangent intersects at two other points.

b.1) Show that the abscissae of these points are the solutions of the equation :

$x^4-8x^2\,+\,12x\,-5\,=\,0$

b.2) Verify that we have :

$x^4\,-8x^2\,+\,12x\,-5\,=\,(x\,-\,1)^2(x^2\,+\,2x\,-\,5)$

b.3) Deduce the abscissae of these points.

Exercise 22 – Parabolas and tangents

Let (P) be the parabola with equation $y=x^2-3x+\frac{5}{4}$

and (H) the hyperbola of equation $y=\frac{3(3x+5)}{4(x+3)}$ .

The plane is reduced to an orthonormal reference frame.

1) Show that (P) and (H) meet the axis (Oy) at the same point A.

2) Show that the tangents at A to curves (P) and (H) are perpendicular.

Reminder: In a n.o.r. two lines are perpendicular if and only if the product of their directing coefficient is equal to -1 .

Exercise 23 – Tangent and determining a real

Determine the real m so that the curve of equation $y\,=\,(m\,-\,1)\,x^2\,+\,(\,3m\,+\,2)\,x\,+\,4$

admits at the point of abscissa -1 a tangent of direction coefficient 6.

Exercise 24 – Determine the abscissa of a tangent

Let be the function $f:x\,\mapsto \,\frac{-x^2\,+2x-1}{x}$ defined on $\mathbb{R}^*$ and let (C) be its representative curve.

Determine the abscissa of the points on (C) where the tangent :

1) is horizontal

2) is parallel to the line of equation $y=-\frac{2}{3}x-5$ .

Exercise 25 – Finding the expression of a square function

A parabola admits in a reference frame $(O;\vec{i},\vec{j})$ an equation of the type :

$y=ax^2+bx+c\,(a\neq0)$
1. Determine the coefficients a, b and c knowing that intersects the x-axis at point A with abscissa 3, the y-axis at point B with ordinate 2 and that it has at this point the line of equation y = 2x + 2 as tangent.

2. Indicate the abscissa of the second point of intersection of with (Ox).

Exercise 26 – Derivative number and tangent to a curve

In each case determine and give an equation of the tangent T.

Exercise 27 – Equation of tangent to a parabola
Consider the function f defined by :
$f(x)\,=\,ax^2\,+\,bx\,+\,c$

whose parabola (Cf ) passes through points A (0; 1) and B (2; 3).

The tangents at A and B intersect at point C (1; – 4).

1. Determine an equation of the tangents to (Cf ).

Deduce f ‘ (0) and f ‘ (2).

2. Express f ‘ (x) as a function of a, b and c.

3. Using the values of f ‘ (0), f ‘ (2) and f(0), find three equations verified by a, b and c and then determine the algebraic expression of the function f.

Exercise 28 – Limit at infinity and table of variation
Consider the function defined on $\mathbb{R}$ by $f(x)=\frac{x}{x^2+1}$ .

1. Calculate the limits of f in $+\infty$ and $-\infty$ .

2. Calculate the derivative f ” of f and study its sign.

3. Draw up the table of variation of the function f.

Exercise 29 – Graphical reading
Below is given the curve (Cf ) representing a function f defined and derivable on the interval [2 ; 7].

1. By graphical reading, give without justification the value of :

f(3); f ‘ (3); f(6); f ‘ (6).

2. The graph does not allow the reading of f ‘ (4).
Nevertheless, specify its sign. Explain.

Exercise 30 – Calculation of a derivative and table of variation
Let be the function defined on $\mathbb{R}$ by $f(x)\,=\,-x^3\,-\,3x^2\,+\,9x$ .

1. Calculate the derivative and study its sign.

2. Draw up the table of variations of the function f.

Exercise 31 – Graphical reading of the derivative number

On the graph below are represented the curve (Cf ) of the function f defined on $\mathbb{R}$ by :

$f(x)=(1-\frac{x}{2})^4$ and the tangent (T) to (Cf ) at the point of abscissa .

1. Give, by graphical reading, and without justifications, the value of the number f ‘ (4).

2. Determine the value of the number f ‘ (3) by calculating the derivative of f.

Exercise 32 – Derivability at a point

Let be the function defined on $\mathbb{R}^*$ by $f(x)=\frac{1}{x}+2$ .

1. Show that f is differentiable in 2.

2. Determine an equation of the tangent (T) to the curve (Cf ) representing f at the point of abscissa 2.

Exercise 33 – Calculation of derivative and derivative number

1. Derive the functions f and g defined below:

$f(x)=\frac{x}{x+\sqrt{x}}\,sur\,%5D0;+\infty%5B$

$g(x)=(\frac{1}{1+x})^3\,sur\,\mathbb{R}-\{-1}$

2. Calculate f ‘ (16) and g ‘ (2).

Exercise 34 – Sense of variation and framing
1. Study the direction of variation of the function defined on $\mathbb{R}$ by :.

2. Deduce a framework of f(x) on [0 ; 2].

Exercise 35 – Study of a numerical function
Consider the function defined on $\mathbb{R}^*$ by $f(x)=x-2+\frac{4}{x}$ .

1. Calculate the derivative f ‘ and study its sign.

2. Draw up the table of variations of the function f.

3. Draw the graphical representation (Cf ) of the function f on $%5B-4\,;\,0%5B\cup\,%5D0\,;\,4%5D$ .

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