Exponential: senior math exercises corrected in PDF.

Corrected math exercises for high school seniors on exponential functions.

These exercises involve the following concepts:

  1. definition of the exponential;
  2. direction of variation of the exponential function;
  3. derivative of the exponential function;
  4. limits of the exponential function;
  5. solve equations and inequalities;
  6. Gauss curve;
  7. simplify exponentials using algebraic formulas.

Exercise #1:

Write using a single exponential :

a. \frac{1}{e^3}

b. e^{-2}\times  \,e^7

Exercise #2:

f is a differentiable function on \mathbb{R} such that f'=f\,\,et\,\,f(0)=-\frac{1}{2}.

g is the function defined on \mathbb{R} by g(x)=-2f(x).

  1. Check that g is derivable on \mathbb{R} and that g’ = g.
  2. Calculate g(0); derive the expression for g(x).
  3. Deduce the expression of f(x).

Exercise #3:

In each case, write the expression with a single exponential.

1.

a. e^4\times  \,e^6

b. e\times  \,(e^5)^2

c. \frac{e^{30}\times  \,e^{-10}}{e^{10}}

2. a is a real number, simplify the writing of each expression :

a)\,\,\frac{e^{2a}\times  \,e^{-a}}{e^{5a}}\,\,;b)\,\,\frac{e^{2a}+1}{e^{1-a}}\,\,;\,\,c)(e^a)^3\times  \,e

Exercise #4:

f is the function defined on by .

In a reference frame, \xi is the representative curve of the function f and T_a is the tangent to \xi at the point A of abscissa a with a>-1.

1. give an equation of T_a.

2. Prove that there are two values of a for which T_a passes through the origin of the frame of reference.

Exercise #5:

We model the average temperature T inside a freezer by posing :

T(t)=19,5e^{-7\times  \,10^{-4}t}-10,5 where t\in%5B0;+\infty%5B is the elapsed time, expressed in minutes

since it was switched on and T(t) its temperature in °C.

1. Give the average temperature inside the freezer:

a. before starting it up;

b. after one day of operation.

2. Study the limit of T in +\infty and interpret the result obtained.

Exercise #6:

Write the given reals in exponential form e^k where k is an integer.

1)e^{-7}\times  \,e^3

2)e^{-1}\times  \,e^{-5}

3)e^2,\times  \,e

4)e\times  \,e^{-1}

5)\frac{1}{e}

6)\frac{1}{e^{-1}}

7)\frac{1}{e^2}

8)\frac{1}{e^{-3}}

9)\frac{e^{-3}}{e^2}

10)\frac{e}{e^{-1}}

11)\frac{e^{-2}}{e}

12)\frac{e^2\times  \,e^{-3}}{e^5}

13)(e^2)^3

14)(e^3)^2

15)(e^{-1})^6

16)e\times  \,(e^{-1})^3

Exercise #6:

Write the given expression in the form e^A where A is an expression.

1)e^x,\times  \,e^2

2)e^{-1}\times  \,e^{-x}

3)e\times  \,e^x

4)e^x\times  \,e^x

5)e^x\times  \,e^{-x}

6)e^{x-1}\times  \,e^x

7)(e^x)^2

8)(e^{-x+1})^3

9)(2e^x)^3

10)\frac{e^{5x}}{e^x}

11)\frac{e^{x+1}}{e}

12)\frac{e^3}{e^{2x-1}}

Exercise #7:

We give the expression of three functions f,g and h defined and derivable on \mathbb{R}.

Calculate the derivative of the functions f, g and h.

1)f(x)=e^{0,5x};g(x)=e^{3x};h(x)=e^{-x}.

2)f(x)=e^{x+1};g(x)=e^{1-2x};h(x)=e^{-3x+1}

3)f(x)=2+e^{2x};g(x)=1-e^{-2x};h(x)=e^{-3x+1}

4)f(x)=2+e^{2x};g(x)=1-e^{-2x};h(x)=1+2e^{-x}

Exercise #8:

It is estimated that future oil discoveries in the world can be modeled,
from 2015, by the function f defined on [15 ; +\infty[ by:

f(x)=17280e^{-0,024x}

where f(x) is the estimated quantity in millions of barrels
of oil that will be discovered in the year 2000 + x.
1. Determine the limit of the function f in +\infty.

2. Calculate f ‘ (x) and deduce the direction of variation of the function f on the interval [15 ; +\infty [.
3. Interpret the results of questions 1 and 2.

exponential

Exercise #9:

Let f be the function defined on \mathbb{R} by f(x)=2-e^x+x.

  1. Express f\,'\,(x) as a function of x.

2) Justify that, for any real x in the interval %5B0;+\infty%5B, f'(x)\leq\,\,0.

3) Deduce the variations of the function f on \mathbb{R}.

Exercise #10:

Write the following expressions in exponential form e^A, where A is an expression.

1)\frac{e^{2x+1}}{e^{1-x}}

2)\frac{e^{-x+2}\times  \,e^{-2x-1}}{e^{3x+2}\times  \,e^{-x-1}}

3)\frac{(e^{-x})^2\times  \,e^{-x+1}}{e^{x+2}\times  \,(e^{-x-1})^3}

Exercise #11:

Prove the following equalities:

For any real x, -2e^{2x}+3e^x+2=(1-2e^x)(2-e^x).

For any real x, \frac{e\times  \,e^x}{e^{2+3x}}=(e^{-x-0,5})^2.

For any real x, \frac{e^{1-3x}}{1+e^{-3x}}=\frac{e}{e^{3x}+1}

Exercise #12:

1)Show that the equation e^x-2e^{-x}+1=0 is equivalent to the equation (e^x)^2+e^x-2=0.

2)Solve in \mathbb{R} the equation e^x-2e^{-x}+1=0.

Exercise #13:

1)Solve in \mathbb{R} the inequation e^{-x}-e^x>0.

2)Deduce the sign of 1-\frac{1+e^x}{1+e^{-x}} on \mathbb{R}.

Exercise #14:

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

and g the function defined on \mathbb{R} by g(x)=\frac{x+1}{e^x}.

The representative curves C_f and C_g of the functions f and g are given below.

  1. Conjecture the limits of the functions f and g at the limits of their definition set.
  2. Prove these conjectures.

exponential

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